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Theorem rexbiia 2427
Description: Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 26-Oct-1999.)
Hypothesis
Ref Expression
ralbiia.1 (𝑥𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rexbiia (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝜓)

Proof of Theorem rexbiia
StepHypRef Expression
1 ralbiia.1 . . 3 (𝑥𝐴 → (𝜑𝜓))
21pm5.32i 449 . 2 ((𝑥𝐴𝜑) ↔ (𝑥𝐴𝜓))
32rexbii2 2423 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wcel 1465  wrex 2394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-ial 1499
This theorem depends on definitions:  df-bi 116  df-rex 2399
This theorem is referenced by:  2rexbiia  2428  ceqsrexbv  2790  reu8  2853  reldm  6052  djur  6922  prarloclem3  7273  suplocexprlemell  7489  recexgt0  8310  fsum3  11124  even2n  11498  lmres  12344
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