http://us2.metamath.org:8888/mpeuni/mmrecent.html Recent proofs for Metamath proof system en http://us2.metamath.org:8888/mpeuni/fopabco.html 21-Jan-2012v may be assigned to w, y, or z if desired. (Unnecessary distinct variable restrictions were removed by David Abernethy, 21-Jan-2012.)
   ⊢ R ∈ V    &
   ⊢ S ∈ V    &
   ⊢ T ∈ V    &
   ⊢ (z = R → S = T)    &
   ⊢ F = {x, y ∣ (x ∈ A ⋀ y = R)}    &
   ⊢ G = {z, w ∣ (z ∈ B ⋀ w = S)}    &
   ⊢ H = {x, v ∣ (x ∈ A ⋀ v = T)}    ⇒
   ⊢ (ran F ⊆ B → (G ∘ F) = H)
    ]]> http://us2.metamath.org:8888/mpeuni/abf.html 20-Jan-2012   ⊢ ¬ φ    ⇒
   ⊢ {x ∣ φ} = ∅
    ]]> http://us2.metamath.org:8888/mpeuni/riotaxfrd.html 16-Jan-2012x in the expression for "the unique A such that φ " to another variable y contained in expression B. Use reuhypd 3660 to eliminate the last hypothesis. (Th. reuunixfr 3662 analog.)
   ⊢ (z ∈ C → ∀y z ∈ C)    &
   ⊢ ((φ ⋀ y ∈ A) → B ∈ A)    &
   ⊢ ((φ ⋀ (ι y ∈ Aχ) ∈ A) → C ∈ A)    &
   ⊢ (x = B → (ψ ↔ χ))    &
   ⊢ (y = (ι y ∈ Aχ) → B = C)    &
   ⊢ ((φ ⋀ x ∈ A) → ∃!y ∈ A x = B)    ⇒
   ⊢ ((φ ⋀ ∃!x ∈ A ψ) → (ι x ∈ Aψ) = C)
    ]]> http://us2.metamath.org:8888/mpeuni/riotacl2.html 16-Jan-2012A such that φ." (Th. reucl2 3626 analog.)
   ⊢ (∃!x ∈ A φ → (ι x ∈ Aφ) ∈ {x ∈ A ∣ φ})
    ]]> http://us2.metamath.org:8888/mpeuni/riotabiia.html 16-Jan-2012rabbiia 2118 analog.)
   ⊢ (x ∈ A → (ψ ↔ χ))    ⇒
   ⊢ (ι x ∈ Aψ) = (ι x ∈ Aχ)
    ]]> http://us2.metamath.org:8888/mpeuni/riotaund.html 16-Jan-2012A when not meaningful.
   ⊢ (¬ ∃!x ∈ A φ → (ι x ∈ Aφ) = (Undef ‘A))
    ]]> http://us2.metamath.org:8888/mpeuni/riotacl.html 16-Jan-2012   ⊢ (∃!x ∈ A φ → (ι x ∈ Aφ) ∈ A)
    ]]> http://us2.metamath.org:8888/mpeuni/reuhypd.html 16-Jan-2012riotaxfrd 5392.
   ⊢ ((φ ⋀ x ∈ C) → B ∈ C)    &
   ⊢ ((φ ⋀ x ∈ C ⋀ y ∈ C) → (x = A ↔ y = B))    ⇒
   ⊢ ((φ ⋀ x ∈ C) → ∃!y ∈ C x = A)
    ]]> http://us2.metamath.org:8888/mpeuni/reuxfrd.html 16-Jan-2012x to another variable y contained in expression A. Use reuhypd 3660 to eliminate the second hypothesis.
   ⊢ ((φ ⋀ y ∈ B) → A ∈ B)    &
   ⊢ ((φ ⋀ x ∈ B) → ∃!y ∈ B x = A)    &
   ⊢ (x = A → (ψ ↔ χ))    ⇒
   ⊢ (φ → (∃!x ∈ B ψ ↔ ∃!y ∈ B χ))
    ]]> http://us2.metamath.org:8888/mpeuni/reuxfr2d.html 16-Jan-2012x to another variable y contained in expression A.
   ⊢ ((φ ⋀ y ∈ B) → A ∈ B)    &
   ⊢ ((φ ⋀ x ∈ B) → ∃*y(y ∈ B ⋀ x = A))    ⇒
   ⊢ (φ → (∃!x ∈ B ∃y ∈ B (x = A ⋀ ψ) ↔ ∃!y ∈ B ψ))]]>