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Theorem rexbid 2373
Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 27-Jun-1998.)
Hypotheses
Ref Expression
ralbid.1 𝑥𝜑
ralbid.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
rexbid (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))

Proof of Theorem rexbid
StepHypRef Expression
1 ralbid.1 . 2 𝑥𝜑
2 ralbid.2 . . 3 (𝜑 → (𝜓𝜒))
32adantr 270 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
41, 3rexbida 2369 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wnf 1390  wcel 1434  wrex 2354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-rex 2359
This theorem is referenced by:  rexbidv  2375  sbcrext  2902  caucvgsrlemgt1  7243  bezout  10780  sscoll2  11226
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