Theorem opabbidv 4625

Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 15-May-1995.)

Hypothesis

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

Assertion

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

Distinct variable groups:   φ,x   φ,y

Allowed substitution hints:   ψ(x,y)   χ(x,y)

Proof of Theorem opabbidv

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎxφ
2		nfv 1619	. 2 ⊢ Ⅎyφ
3		opabbidv.1	. 2 ⊢ (φ → (ψ ↔ χ))
4	1, 2, 3	opabbid 4624	1 ⊢ (φ → {⟨x, y⟩ ∣ ψ} = {⟨x, y⟩ ∣ χ})

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  {copab 4622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-opab 4623

This theorem is referenced by:  opabbii  4626  csbopabg  4637  xpeq1  4798  xpeq2  4799  opabbi2dv  4867  resopab2  5001  cores  5084  f1oiso2  5500  f1od  5726