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Theorem exbid 1578
Description: Formula-building rule for existential quantifier (deduction form). (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 1482 . 2 (𝜑 → ∀𝑥𝜑)
3 exbid.2 . 2 (𝜑 → (𝜓𝜒))
42, 3exbidh 1576 1 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wnf 1419  wex 1451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-ial 1497
This theorem depends on definitions:  df-bi 116  df-nf 1420
This theorem is referenced by:  mobid  2010  rexbida  2407  rexeqf  2598  opabbid  3961  repizf2  4054  oprabbid  5790  sscoll2  12997
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