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Theorem exbidh 1521
 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 1374 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 exbi 1511 . 2 (∀𝑥(𝜓𝜒) → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
53, 4syl 14 1 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102  ∀wal 1257  ∃wex 1397 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443 This theorem depends on definitions:  df-bi 114 This theorem is referenced by:  exbid  1523  drex2  1636  drex1  1695  exbidv  1722  mobidh  1950
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