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Theorem exbid 1616
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 1519 . 2 (𝜑 → ∀𝑥𝜑)
3 exbid.2 . 2 (𝜑 → (𝜓𝜒))
42, 3exbidh 1614 1 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wnf 1460  wex 1492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-nf 1461
This theorem is referenced by:  mobid  2061  rexbida  2472  rexbid2  2482  rexeqf  2669  opabbid  4068  repizf2  4162  oprabbid  5927
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