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Theorem rexbiOLD 3173
Description: Obsolete version of rexbi 3172 as of 31-Oct-2024. (Contributed by Scott Fenton, 7-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
rexbiOLD (∀𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝜓))

Proof of Theorem rexbiOLD
StepHypRef Expression
1 ralbi 3095 . . 3 (∀𝑥𝐴𝜑 ↔ ¬ 𝜓) → (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥𝐴 ¬ 𝜓))
21notbid 318 . 2 (∀𝑥𝐴𝜑 ↔ ¬ 𝜓) → (¬ ∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜓))
3 notbi 319 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))
43ralbii 3093 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴𝜑 ↔ ¬ 𝜓))
5 dfrex2 3169 . . 3 (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑)
6 dfrex2 3169 . . 3 (∃𝑥𝐴 𝜓 ↔ ¬ ∀𝑥𝐴 ¬ 𝜓)
75, 6bibi12i 340 . 2 ((∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝜓) ↔ (¬ ∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜓))
82, 4, 73imtr4i 292 1 (∀𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wral 3066  wrex 3067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1787  df-ral 3071  df-rex 3072
This theorem is referenced by: (None)
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