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Theorem rexbiia 2450
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 2446 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wcel 1480  wrex 2417
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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-rex 2422
This theorem is referenced by:  2rexbiia  2451  ceqsrexbv  2816  reu8  2880  reldm  6084  djur  6954  prarloclem3  7312  suplocexprlemell  7528  recexgt0  8349  fsum3  11163  even2n  11578  lmres  12427  reeff1o  12872  ioocosf1o  12948
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