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Theorem exbidh 1505
Description: Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
exbidh.1 (𝜑 → ∀𝑥𝜑)
exbidh.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
exbidh (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Proof of Theorem exbidh
StepHypRef Expression
1 exbidh.1 . . 3 (𝜑 → ∀𝑥𝜑)
2 exbidh.2 . . 3 (𝜑 → (𝜓𝜒))
31, 2alrimih 1358 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 exbi 1495 . 2 (∀𝑥(𝜓𝜒) → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
53, 4syl 14 1 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1241  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
This theorem is referenced by:  exbid  1507  drex2  1620  drex1  1679  exbidv  1706  mobidh  1934
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