Theorem cbvex2v 976

Description: Rule used to change bound variables with implicit substitution.

Hypothesis

Ref	Expression
cbvex2v.1	⊢ ((x = z ∧ y = w) → (φ ↔ ψ))

Assertion

Ref	Expression
cbvex2v	⊢ (∃x∃yφ ↔ ∃z∃wψ)

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

Proof of Theorem cbvex2v

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		cbvex2v.1	. 2 ⊢ ((x = z ∧ y = w) → (φ ↔ ψ))
6	1, 2, 3, 4, 5	cbvex2 975	1 ⊢ (∃x∃yφ ↔ ∃z∃wψ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = weq 797

This theorem is referenced by:  cbvex4v 979  th3qlem1 3250  genpv 3896

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-12 802  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679

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