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Theorem exlimdh 2145
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 2021 . 2 𝑥𝜑
3 exlimdh.2 . . 3 (𝜒 → ∀𝑥𝜒)
43nf5i 2021 . 2 𝑥𝜒
5 exlimdh.3 . 2 (𝜑 → (𝜓𝜒))
62, 4, 5exlimd 2085 1 (𝜑 → (∃𝑥𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-ex 1702  df-nf 1707
This theorem is referenced by:  exlimexi  38251  eexinst01  38253  eexinst11  38254
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