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Theorem rexbid 3235
Description: Formula-building rule for restricted existential quantifier (deduction rule). (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 468 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
41, 3rexbida 3231 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wnf 1863  wcel 2155  wrex 3093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-12 2213
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1860  df-nf 1864  df-rex 3098
This theorem is referenced by:  rexbidvALT  3237  rexeqbid  3336  scott0  8990  infcvgaux1i  14805  bnj1463  31440  poimirlem25  33741  poimirlem26  33742  elrnmptf  39850  smfsupmpt  41497  smfinfmpt  41501
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