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Theorem rexbid 3313
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 3311 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wnf 1785  wcel 2115  wrex 3134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-rex 3139
This theorem is referenced by:  rexbidvALT  3314  rexeqbid  3408  scott0  9306  infcvgaux1i  15208  bnj1463  32352  fvineqsneq  34741  poimirlem25  34992  poimirlem26  34993  elrnmptf  41669  smfsupmpt  43309  smfinfmpt  43313
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