Theorem cbvalv 2370

Description: Rule used to change bound variables, using implicit substitution. See cbvalvw 2086 for a version requiring fewer axioms, to be preferred when sufficient. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-10 2135. (Revised by Wolf Lammen, 17-Jul-2021.)

Hypothesis

Ref	Expression
cbvalv.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvalv	⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥

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

Proof of Theorem cbvalv

Step	Hyp	Ref	Expression
1		ax-5 1953	. . . 4 ⊢ (𝜑 → ∀𝑦𝜑)
2	1	hbal 2160	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
3		cbvalv.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	3	spv 2358	. . 3 ⊢ (∀𝑥𝜑 → 𝜓)
5	2, 4	alrimih 1867	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
6		ax-5 1953	. . . 4 ⊢ (𝜓 → ∀𝑥𝜓)
7	6	hbal 2160	. . 3 ⊢ (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓)
8	3	equcoms 2067	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))
9	8	bicomd 215	. . . 4 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑))
10	9	spv 2358	. . 3 ⊢ (∀𝑦𝜓 → 𝜑)
11	7, 10	alrimih 1867	. 2 ⊢ (∀𝑦𝜓 → ∀𝑥𝜑)
12	5, 11	impbii 201	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-11 2150  ax-12 2163  ax-13 2334

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-nf 1828

This theorem is referenced by:  cbvexv  2371  cbvaldva  2376  cbval2v  2378  cdeqal1  3643