Theorem oprabbidv 5565

Description: Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.)

Hypothesis

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

Assertion

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

Distinct variable groups:   x,z,φ   y,z,φ

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

Proof of Theorem oprabbidv

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎxφ
2		nfv 1619	. 2 ⊢ Ⅎyφ
3		nfv 1619	. 2 ⊢ Ⅎzφ
4		oprabbidv.1	. 2 ⊢ (φ → (ψ ↔ χ))
5	1, 2, 3, 4	oprabbid 5564	1 ⊢ (φ → {⟨⟨x, y⟩, z⟩ ∣ ψ} = {⟨⟨x, y⟩, z⟩ ∣ χ})

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  {coprab 5528

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-oprab 5529

This theorem is referenced by:  oprabbii  5566  resoprab2  5583  mpt2eq123dv  5664  mpt2eq3dva  5670