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Theorem rexeqbidvvOLD 3322
Description: Obsolete version of rexeqbidvv 3321 as of 9-Mar-2025. (Contributed by Wolf Lammen, 25-Sep-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
raleqbidvv.1 (𝜑𝐴 = 𝐵)
raleqbidvv.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
rexeqbidvvOLD (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem rexeqbidvvOLD
StepHypRef Expression
1 raleqbidvv.1 . . . 4 (𝜑𝐴 = 𝐵)
2 raleqbidvv.2 . . . . 5 (𝜑 → (𝜓𝜒))
32notbid 317 . . . 4 (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))
41, 3raleqbidvv 3319 . . 3 (𝜑 → (∀𝑥𝐴 ¬ 𝜓 ↔ ∀𝑥𝐵 ¬ 𝜒))
5 ralnex 3062 . . 3 (∀𝑥𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥𝐴 𝜓)
6 ralnex 3062 . . 3 (∀𝑥𝐵 ¬ 𝜒 ↔ ¬ ∃𝑥𝐵 𝜒)
74, 5, 63bitr3g 312 . 2 (𝜑 → (¬ ∃𝑥𝐴 𝜓 ↔ ¬ ∃𝑥𝐵 𝜒))
87con4bid 316 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1533  wral 3051  wrex 3060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-cleq 2717  df-ral 3052  df-rex 3061
This theorem is referenced by: (None)
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