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Theorem exlimdh 1576
 Description: Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jan-1997.)
Hypotheses
Ref Expression
exlimdh.1 (𝜑 → ∀𝑥𝜑)
exlimdh.2 (𝜒 → ∀𝑥𝜒)
exlimdh.3 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
exlimdh (𝜑 → (∃𝑥𝜓𝜒))

Proof of Theorem exlimdh
StepHypRef Expression
1 exlimdh.1 . . 3 (𝜑 → ∀𝑥𝜑)
2 exlimdh.3 . . 3 (𝜑 → (𝜓𝜒))
31, 2alrimih 1449 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 exlimdh.2 . . 3 (𝜒 → ∀𝑥𝜒)
5419.23h 1478 . 2 (∀𝑥(𝜓𝜒) ↔ (∃𝑥𝜓𝜒))
63, 5sylib 121 1 (𝜑 → (∃𝑥𝜓𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1333  ∃wex 1472 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-5 1427  ax-gen 1429  ax-ie2 1474 This theorem depends on definitions:  df-bi 116 This theorem is referenced by:  exlimd  1577  exim  1579  exlimdv  1799  equs5  1809  cbvexdh  1906  exists2  2103
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