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Theorem exbid 1507
 Description: Formula-building rule for existential quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
Hypotheses
Ref Expression
exbid.1 𝑥𝜑
exbid.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
exbid (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Proof of Theorem exbid
StepHypRef Expression
1 exbid.1 . . 3 𝑥𝜑
21nfri 1412 . 2 (𝜑 → ∀𝑥𝜑)
3 exbid.2 . 2 (𝜑 → (𝜓𝜒))
42, 3exbidh 1505 1 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  Ⅎwnf 1349  ∃wex 1381 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-nf 1350 This theorem is referenced by:  mobid  1935  rexbida  2321  rexeqf  2502  opabbid  3822  repizf2  3915  oprabbid  5558  sscoll2  10113
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