Theorem cbval 1728

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 1500	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
3		cbval.2	. . 3 ⊢ Ⅎ𝑥𝜓
4	3	nfri 1500	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
5		cbval.3	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	2, 4, 5	cbvalh 1727	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1330  Ⅎwnf 1437

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515

This theorem depends on definitions:  df-bi 116  df-nf 1438

This theorem is referenced by:  sb8  1829  cbval2  1894  sb8eu  2013  abbi  2254  cleqf  2306  cbvralf  2651  ralab2  2852  cbvralcsf  3067  dfss2f  3093  elintab  3790  cbviota  5101  sb8iota  5103  dffun6f  5144  dffun4f  5147  mptfvex  5514  findcard2  6791  findcard2s  6792