Theorem cbvalh 1753

Description: Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypotheses

Ref	Expression
cbvalh.1	⊢ (𝜑 → ∀𝑦𝜑)
cbvalh.2	⊢ (𝜓 → ∀𝑥𝜓)
cbvalh.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvalh	⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Proof of Theorem cbvalh

Step	Hyp	Ref	Expression
1		cbvalh.1	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
2		cbvalh.2	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
3		cbvalh.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	3	biimpd 144	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
5	1, 2, 4	cbv3h 1743	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
6	3	equcoms 1708	. . . 4 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))
7	6	biimprd 158	. . 3 ⊢ (𝑦 = 𝑥 → (𝜓 → 𝜑))
8	2, 1, 7	cbv3h 1743	. 2 ⊢ (∀𝑦𝜓 → ∀𝑥𝜑)
9	5, 8	impbii 126	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1351

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-nf 1461

This theorem is referenced by:  cbval  1754  sb8h  1854  cbvalv  1917  sb9v  1978  sb8euh  2049