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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 → (𝜑𝜓))
Assertion
Ref Expression
cbvalv (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvalv
StepHypRef Expression
1 ax-5 1953 . . . 4 (𝜑 → ∀𝑦𝜑)
21hbal 2160 . . 3 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
3 cbvalv.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
43spv 2358 . . 3 (∀𝑥𝜑𝜓)
52, 4alrimih 1867 . 2 (∀𝑥𝜑 → ∀𝑦𝜓)
6 ax-5 1953 . . . 4 (𝜓 → ∀𝑥𝜓)
76hbal 2160 . . 3 (∀𝑦𝜓 → ∀𝑥𝑦𝜓)
83equcoms 2067 . . . . 5 (𝑦 = 𝑥 → (𝜑𝜓))
98bicomd 215 . . . 4 (𝑦 = 𝑥 → (𝜓𝜑))
109spv 2358 . . 3 (∀𝑦𝜓𝜑)
117, 10alrimih 1867 . 2 (∀𝑦𝜓 → ∀𝑥𝜑)
125, 11impbii 201 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
 Colors of variables: wff setvar class 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
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