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Theorem rexbid 3239
Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 27-Jun-1998.)
Hypotheses
Ref Expression
rexbid.1 𝑥𝜑
rexbid.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
rexbid (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))

Proof of Theorem rexbid
StepHypRef Expression
1 rexbid.1 . 2 𝑥𝜑
2 rexbid.2 . . 3 (𝜑 → (𝜓𝜒))
32adantr 484 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
41, 3rexbida 3237 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wnf 1791  wcel 2110  wrex 3062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-12 2175
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-nf 1792  df-rex 3067
This theorem is referenced by:  rexbidvALT  3240  rexeqbid  3330  scott0  9502  infcvgaux1i  15421  bnj1463  32748  fvineqsneq  35320  poimirlem25  35539  poimirlem26  35540  elrnmptf  42394  smfsupmpt  44023  smfinfmpt  44027
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