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Theorem exlimddvf 34412
Description: A lemma for eliminating an existential quantifier. (Contributed by Giovanni Mascellani, 30-May-2019.)
Hypotheses
Ref Expression
exlimddvf.1 (𝜑 → ∃𝑥𝜃)
exlimddvf.2 𝑥𝜓
exlimddvf.3 ((𝜃𝜓) → 𝜒)
exlimddvf.4 𝑥𝜒
Assertion
Ref Expression
exlimddvf ((𝜑𝜓) → 𝜒)

Proof of Theorem exlimddvf
StepHypRef Expression
1 exlimddvf.1 . 2 (𝜑 → ∃𝑥𝜃)
2 exlimddvf.2 . . 3 𝑥𝜓
3 exlimddvf.4 . . 3 𝑥𝜒
4 exlimddvf.3 . . . 4 ((𝜃𝜓) → 𝜒)
54expcom 403 . . 3 (𝜓 → (𝜃𝜒))
62, 3, 5exlimd 2253 . 2 (𝜓 → (∃𝑥𝜃𝜒))
71, 6mpan9 503 1 ((𝜑𝜓) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wex 1875  wnf 1879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-12 2213
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-nf 1880
This theorem is referenced by:  exlimddvfi  34413
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