Theorem cbvalv1 1731

Description: Rule used to change bound variables, using implicit substitution. Version of cbval 1734 with a disjoint variable condition. See cbvalvw 1899 for a version with two disjoint variable conditions, and cbvalv 1897 for another variant. (Contributed by NM, 13-May-1993.) (Revised by BJ, 31-May-2019.)

Hypotheses

Ref	Expression
cbvalv1.nf1	⊢ Ⅎ𝑦𝜑
cbvalv1.nf2	⊢ Ⅎ𝑥𝜓
cbvalv1.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvalv1	⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvalv1

Step	Hyp	Ref	Expression
1		cbvalv1.nf1	. . 3 ⊢ Ⅎ𝑦𝜑
2		cbvalv1.nf2	. . 3 ⊢ Ⅎ𝑥𝜓
3		cbvalv1.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	3	biimpd 143	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
5	1, 2, 4	cbv3v 1724	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
6	3	biimprd 157	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑))
7	6	equcoms 1688	. . 3 ⊢ (𝑦 = 𝑥 → (𝜓 → 𝜑))
8	2, 1, 7	cbv3v 1724	. 2 ⊢ (∀𝑦𝜓 → ∀𝑥𝜑)
9	5, 8	impbii 125	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1333  Ⅎwnf 1440

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-nf 1441

This theorem is referenced by: (None)