Theorem cbvalv 1835

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 1459	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
2		ax-17 1459	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
3		cbvalv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvalh 1676	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 103  ∀wal 1282

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-nf 1390

This theorem is referenced by:  nfcjust  2207  cdeqal1  2806  zfpow  3949  tfisi  4328  acexmid  5531  tfrlem3-2d  5951  tfrlemi1  5969  tfrexlem  5971  findcard  6372  genprndl  6711  genprndu  6712