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Theorem rexbid 3317
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 481 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
41, 3rexbida 3315 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wnf 1775  wcel 2105  wrex 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-nf 1776  df-rex 3141
This theorem is referenced by:  rexbidvALT  3318  rexeqbid  3420  scott0  9303  infcvgaux1i  15200  bnj1463  32224  fvineqsneq  34575  poimirlem25  34798  poimirlem26  34799  elrnmptf  41317  smfsupmpt  42966  smfinfmpt  42970
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