Theorem cbval 2412

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. Check out cbvalw 2038, cbvalvw 2039, cbvalv1 2357 for versions requiring fewer axioms. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (New usage is discouraged.)

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		cbval.2	. . 3 ⊢ Ⅎ𝑥𝜓
3		cbval.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	3	biimpd 231	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
5	1, 2, 4	cbv3 2411	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
6	3	biimprd 250	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑))
7	6	equcoms 2023	. . 3 ⊢ (𝑦 = 𝑥 → (𝜓 → 𝜑))
8	2, 1, 7	cbv3 2411	. 2 ⊢ (∀𝑦𝜓 → ∀𝑥𝜑)
9	5, 8	impbii 211	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1531  Ⅎwnf 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-11 2157  ax-12 2173  ax-13 2386

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-nf 1781

This theorem is referenced by:  cbvex  2413  cbvalv  2414  cbval2  2428  cbval2OLD  2429  sb8  2555