Theorem biopabdv 2102

Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction rule).

Hypothesis

Ref	Expression
biopabdv.1	⊢ (φ → (ψ ↔ χ))

Assertion

Ref	Expression
biopabdv	⊢ (φ → {⟨x, y⟩∣ψ} = {⟨x, y⟩∣χ})

Distinct variable group(s):   x,y,φ

Proof of Theorem biopabdv

Step	Hyp	Ref	ExprÚssion
1		ax-17 925	. 2 ⊢ (φ → ∀xφ)
2		ax-17 925	. 2 ⊢ (φ → ∀yφ)
3		biopabdv.1	. 2 ⊢ (φ → (ψ ↔ χ))
4	1, 2, 3	biopabd 2101	1 ⊢ (φ → {⟨x, y⟩∣ψ} = {⟨x, y⟩∣χ})

Syntax hints:   → wi 2   ↔ wb 127   = wceq 1091  {copab 2055

This theorem is referenced by:  biopabi 2103  xpeq1 2440  xpeq2 2441  coeq1 2502  coeq2 2503  rdgeq1 2972  rdgeq2 2973  omv 3120  oev 3122  en2d 3303  unfilem3 3440  seqval 4665  occllem5 5184  pjmvalt 5245  hosmvalt 5487  hodmvalt 5488

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-opab 2098

metamath.org