Theorem cbvalv 1973

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 1579	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
2		ax-17 1579	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
3		cbvalv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvalh 1806	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 105  ∀wal 1400

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-nf 1514

This theorem is used by:  nfcjust  2380  cdeqal1  3042  dfss4st  3464  zfpow  4312  tfisi  4734  acexmid  6084  tfrlem3-2d  6583  tfrlemi1  6603  tfrexlem  6605  tfr1onlemaccex  6619  tfrcllemaccex  6632  findcard  7192  fisseneq  7242  genprndl  7888  genprndu  7889  zfz1iso  11295