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Theorem exlimdh 1503
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 1374 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 exlimdh.2 . . 3 (𝜒 → ∀𝑥𝜒)
5419.23h 1403 . 2 (∀𝑥(𝜓𝜒) ↔ (∃𝑥𝜓𝜒))
63, 5sylib 131 1 (𝜑 → (∃𝑥𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1257  wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-5 1352  ax-gen 1354  ax-ie2 1399
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  exlimd  1504  exim  1506  exlimdv  1716  equs5  1726  cbvexdh  1817  exists2  2013
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