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Theorem bj-cbval 33149
 Description: Changing a bound variable (universal quantification case) in a weak axiomatization, assuming that all variables denote (which is valid in inclusive free logic) and that equality is symmetric. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-cbval.denote 𝑦𝑥 𝑥 = 𝑦
bj-cbval.denote2 𝑥𝑦 𝑦 = 𝑥
bj-cbval.maj (𝑥 = 𝑦 → (𝜑𝜓))
bj-cbval.equcomiv (𝑦 = 𝑥𝑥 = 𝑦)
Assertion
Ref Expression
bj-cbval (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem bj-cbval
StepHypRef Expression
1 bj-cbval.maj . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
21biimpd 221 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
3 bj-cbval.denote . . 3 𝑦𝑥 𝑥 = 𝑦
42, 3bj-cbvalimi 33147 . 2 (∀𝑥𝜑 → ∀𝑦𝜓)
5 bj-cbval.equcomiv . . . 4 (𝑦 = 𝑥𝑥 = 𝑦)
61biimprd 240 . . . 4 (𝑥 = 𝑦 → (𝜓𝜑))
75, 6syl 17 . . 3 (𝑦 = 𝑥 → (𝜓𝜑))
8 bj-cbval.denote2 . . 3 𝑥𝑦 𝑦 = 𝑥
97, 8bj-cbvalimi 33147 . 2 (∀𝑦𝜓 → ∀𝑥𝜑)
104, 9impbii 201 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1654  ∃wex 1878 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009 This theorem depends on definitions:  df-bi 199  df-ex 1879 This theorem is referenced by: (None)
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