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Theorem rexeqbidvv 3308
Description: Version of rexeqbidv 3306 with additional disjoint variable conditions, not requiring ax-8 2116 nor df-clel 2812. (Contributed by Wolf Lammen, 25-Sep-2024.)
Hypotheses
Ref Expression
raleqbidvv.1 (𝜑𝐴 = 𝐵)
raleqbidvv.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
rexeqbidvv (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem rexeqbidvv
StepHypRef Expression
1 raleqbidvv.1 . . . 4 (𝜑𝐴 = 𝐵)
2 raleqbidvv.2 . . . . 5 (𝜑 → (𝜓𝜒))
32notbid 321 . . . 4 (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))
41, 3raleqbidvv 3307 . . 3 (𝜑 → (∀𝑥𝐴 ¬ 𝜓 ↔ ∀𝑥𝐵 ¬ 𝜒))
5 ralnex 3150 . . 3 (∀𝑥𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥𝐴 𝜓)
6 ralnex 3150 . . 3 (∀𝑥𝐵 ¬ 𝜒 ↔ ¬ ∃𝑥𝐵 𝜒)
74, 5, 63bitr3g 316 . 2 (𝜑 → (¬ ∃𝑥𝐴 𝜓 ↔ ¬ ∃𝑥𝐵 𝜒))
87con4bid 320 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209   = wceq 1542  wral 3054  wrex 3055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787  df-cleq 2731  df-ral 3059  df-rex 3060
This theorem is referenced by:  rexeqbi1dv  3310
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