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Theorem rexbid 3261
Description: Formula-building rule for restricted existential quantifier (deduction form). For a version based on fewer axioms see rexbidv 3168. (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 479 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
41, 3rexbida 3259 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wnf 1777  wcel 2098  wrex 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2166
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-nf 1778  df-rex 3060
This theorem is referenced by:  rexbidvALT  3262  rexeqbid  3340  scott0  9925  infcvgaux1i  15856  bnj1463  34856  fvineqsneq  37067  poimirlem25  37294  poimirlem26  37295  elrnmptf  44725  smfsupmpt  46373  smfinfmpt  46377
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