Theorem cbvalv 1772

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 1396	. 2 ⊢ (φ → ∀yφ)
2		ax-17 1396	. 2 ⊢ (ψ → ∀xψ)
3		cbvalv.1	. 2 ⊢ (x = y → (φ ↔ ψ))
4	1, 2, 3	cbvalh 1614	1 ⊢ (∀xφ ↔ ∀yψ)

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1224

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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405

This theorem depends on definitions:  df-bi 110  df-nf 1326

This theorem is referenced by:  nfcjust  2144  cdeqal1  2728  zfpow  3898  tfisi  4233  acexmid  5431  tfrlem3-2d  5846  tfrlemi1  5863  tfrexlem  5866  genprndl  6370  genprndu  6371