Theorem cbval 1747

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 1512	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
3		cbval.2	. . 3 ⊢ Ⅎ𝑥𝜓
4	3	nfri 1512	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
5		cbval.3	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	2, 4, 5	cbvalh 1746	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1346  Ⅎwnf 1453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by:  sb8  1849  cbval2  1914  sb8eu  2032  abbi  2284  cleqf  2337  cbvralf  2689  ralab2  2894  cbvralcsf  3111  dfss2f  3138  elintab  3842  cbviota  5165  sb8iota  5167  dffun6f  5211  dffun4f  5214  mptfvex  5581  findcard2  6867  findcard2s  6868