Theorem cbvalh 1746

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 143	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
5	1, 2, 4	cbv3h 1736	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
6	3	equcoms 1701	. . . 4 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))
7	6	biimprd 157	. . 3 ⊢ (𝑦 = 𝑥 → (𝜓 → 𝜑))
8	2, 1, 7	cbv3h 1736	. 2 ⊢ (∀𝑦𝜓 → ∀𝑥𝜑)
9	5, 8	impbii 125	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

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

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:  cbval  1747  sb8h  1847  cbvalv  1910  sb9v  1971  sb8euh  2042