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Theorem rexbiia 2339
 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 427 . 2 ((𝑥𝐴𝜑) ↔ (𝑥𝐴𝜓))
32rexbii2 2335 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∈ wcel 1393  ∃wrex 2307 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-rex 2312 This theorem is referenced by:  2rexbiia  2340  ceqsrexbv  2675  reu8  2737  reldm  5812  prarloclem3  6595  recexgt0  7571
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