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Theorem exlimdh 2289
Description: Deduction form of Theorem 19.9 of [Margaris] p. 89. (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)
21nf5i 2141 . 2 𝑥𝜑
3 exlimdh.2 . . 3 (𝜒 → ∀𝑥𝜒)
43nf5i 2141 . 2 𝑥𝜒
5 exlimdh.3 . 2 (𝜑 → (𝜓𝜒))
62, 4, 5exlimd 2208 1 (𝜑 → (∃𝑥𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1526  wex 1771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-ex 1772  df-nf 1776
This theorem is referenced by:  exlimexi  40735  eexinst01  40737  eexinst11  40738
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