Theorem cbval 1765

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

Hypotheses

Ref	Expression
cbval.1	⊢ Ⅎ𝑦𝜑
cbval.2	⊢ Ⅎ𝑥𝜓
cbval.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbval	⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Proof of Theorem cbval

Step	Hyp	Ref	Expression
1		cbval.1	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	nfri 1530	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
3		cbval.2	. . 3 ⊢ Ⅎ𝑥𝜓
4	3	nfri 1530	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
5		cbval.3	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	2, 4, 5	cbvalh 1764	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362  Ⅎwnf 1471

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472

This theorem is referenced by:  sb8  1867  cbval2  1933  sb8eu  2051  abbi  2303  cleqf  2357  cbvralf  2710  ralab2  2916  cbvralcsf  3134  dfss2f  3161  elintab  3873  cbviota  5204  sb8iota  5206  dffun6f  5251  dffun4f  5254  mptfvex  5625  findcard2  6921  findcard2s  6922