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Theorem exbid 1550
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 1455 . 2 (𝜑 → ∀𝑥𝜑)
3 exbid.2 . 2 (𝜑 → (𝜓𝜒))
42, 3exbidh 1548 1 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wnf 1392  wex 1424
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 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-nf 1393
This theorem is referenced by:  mobid  1980  rexbida  2371  rexeqf  2555  opabbid  3878  repizf2  3972  oprabbid  5659  sscoll2  11321
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