Theorem cbvopabv 2105

Description: Rule used to change bound variables in an ordered pair abstraction, using implicit substitution.

Hypothesis

Ref	Expression
cbvopabv.1	⊢ ((x = z ∧ y = w) → (φ ↔ ψ))

Assertion

Ref	Expression
cbvopabv	⊢ {⟨x, y⟩∣φ} = {⟨z, w⟩∣ψ}

Distinct variable group(s):   x,y,z,w   φ,z,w   ψ,x,y

Proof of Theorem cbvopabv

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 ⊢ (φ → ∀zφ)
2		ax-17 925	. 2 ⊢ (φ → ∀wφ)
3		ax-17 925	. 2 ⊢ (ψ → ∀xψ)
4		ax-17 925	. 2 ⊢ (ψ → ∀yψ)
5		cbvopabv.1	. 2 ⊢ ((x = z ∧ y = w) → (φ ↔ ψ))
6	1, 2, 3, 4, 5	cbvopab 2104	1 ⊢ {⟨x, y⟩∣φ} = {⟨z, w⟩∣ψ}

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = weq 797   = wceq 1091  {copab 2055

This theorem is referenced by:  inf3lema 3460  numth 3599  zorn 3611  seqsuclem 4669

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-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098

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