Theorem cbval2v 2006

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.)

Hypothesis

Ref	Expression
cbval2v.1	⊢ ((x = z ∧ y = w) → (φ ↔ ψ))

Assertion

Ref	Expression
cbval2v	⊢ (∀x∀yφ ↔ ∀z∀wψ)

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

Allowed substitution hints:   φ(x,y)   ψ(z,w)

Proof of Theorem cbval2v

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎzφ
2		nfv 1619	. 2 ⊢ Ⅎwφ
3		nfv 1619	. 2 ⊢ Ⅎxψ
4		nfv 1619	. 2 ⊢ Ⅎyψ
5		cbval2v.1	. 2 ⊢ ((x = z ∧ y = w) → (φ ↔ ψ))
6	1, 2, 3, 4, 5	cbval2 2004	1 ⊢ (∀x∀yφ ↔ ∀z∀wψ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540

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

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  nnsucelr  4429  ssfin  4471  fnfrec  6321