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Theorem rexbi 2446
 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 2355 . 2 𝑥𝑥𝐴 (𝜑𝜓)
2 rsp 2369 . . 3 (∀𝑥𝐴 (𝜑𝜓) → (𝑥𝐴 → (𝜑𝜓)))
32imp 115 . 2 ((∀𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜑𝜓))
41, 3rexbida 2321 1 (∀𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∈ wcel 1393  ∀wral 2306  ∃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-nf 1350  df-ral 2311  df-rex 2312 This theorem is referenced by:  rexrnmpt2  5616
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