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Theorem exlimexi 37545
Description: Inference similar to Theorem 19.23 of [Margaris] p. 90. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
exlimexi.1 (𝜓 → ∀𝑥𝜓)
exlimexi.2 (∃𝑥𝜑 → (𝜑𝜓))
Assertion
Ref Expression
exlimexi (∃𝑥𝜑𝜓)

Proof of Theorem exlimexi
StepHypRef Expression
1 hbe1 2008 . . 3 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
2 exlimexi.1 . . 3 (𝜓 → ∀𝑥𝜓)
3 exlimexi.2 . . 3 (∃𝑥𝜑 → (𝜑𝜓))
41, 2, 3exlimdh 2134 . 2 (∃𝑥𝜑 → (∃𝑥𝜑𝜓))
54pm2.43i 50 1 (∃𝑥𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1473  wex 1695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034
This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701
This theorem is referenced by:  sb5ALT  37546  exinst  37664
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