Theorem cbvalv 1791

Description: Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
cbvalv.1	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
cbvalv	⊢ (∀xφ ↔ ∀yψ)

Distinct variable groups:   φ,y   ψ,x

Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem cbvalv

Step	Hyp	Ref	Expression
1		ax-17 1416	. 2 ⊢ (φ → ∀yφ)
2		ax-17 1416	. 2 ⊢ (ψ → ∀xψ)
3		cbvalv.1	. 2 ⊢ (x = y → (φ ↔ ψ))
4	1, 2, 3	cbvalh 1633	1 ⊢ (∀xφ ↔ ∀yψ)

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-nf 1347

This theorem is referenced by:  nfcjust  2163  cdeqal1  2749  zfpow  3919  tfisi  4253  acexmid  5454  tfrlem3-2d  5869  tfrlemi1  5887  tfrexlem  5889  genprndl  6503  genprndu  6504