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Theorem rexbi 2568
Description: Distribute a restricted existential quantifier over a biconditional. Theorem 19.18 of [Margaris] p. 90 with restricted quantification. (Contributed by Jim Kingdon, 21-Jan-2019.)
Assertion
Ref Expression
rexbi (∀𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝜓))

Proof of Theorem rexbi
StepHypRef Expression
1 nfra1 2469 . 2 𝑥𝑥𝐴 (𝜑𝜓)
2 rsp 2483 . . 3 (∀𝑥𝐴 (𝜑𝜓) → (𝑥𝐴 → (𝜑𝜓)))
32imp 123 . 2 ((∀𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜑𝜓))
41, 3rexbida 2433 1 (∀𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wcel 1481  wral 2417  wrex 2418
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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-ial 1515
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-ral 2422  df-rex 2423
This theorem is referenced by:  rexrnmpo  5893
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