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Theorem rexbid 3033
 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 480 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
41, 3rexbida 3029 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  Ⅎwnf 1699   ∈ wcel 1977  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701  df-rex 2902 This theorem is referenced by:  rexbidvALT  3035  rexeqbid  3128  scott0  8632  infcvgaux1i  14428  bnj1463  30377  poimirlem25  32604  poimirlem26  32605  elrnmptf  38362
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