Theorem cbvalv 1967

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 1575	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
2		ax-17 1575	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
3		cbvalv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvalh 1802	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-nf 1510

This theorem is referenced by:  nfcjust  2372  cdeqal1  3032  dfss4st  3453  zfpow  4287  tfisi  4708  acexmid  6048  tfrlem3-2d  6542  tfrlemi1  6562  tfrexlem  6564  tfr1onlemaccex  6578  tfrcllemaccex  6591  findcard  7144  fisseneq  7194  genprndl  7835  genprndu  7836  zfz1iso  11209