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Theorem exlimdd 2087
Description: Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
Hypotheses
Ref Expression
exlimdd.1 𝑥𝜑
exlimdd.2 𝑥𝜒
exlimdd.3 (𝜑 → ∃𝑥𝜓)
exlimdd.4 ((𝜑𝜓) → 𝜒)
Assertion
Ref Expression
exlimdd (𝜑𝜒)

Proof of Theorem exlimdd
StepHypRef Expression
1 exlimdd.3 . 2 (𝜑 → ∃𝑥𝜓)
2 exlimdd.1 . . 3 𝑥𝜑
3 exlimdd.2 . . 3 𝑥𝜒
4 exlimdd.4 . . . 4 ((𝜑𝜓) → 𝜒)
54ex 450 . . 3 (𝜑 → (𝜓𝜒))
62, 3, 5exlimd 2086 . 2 (𝜑 → (∃𝑥𝜓𝜒))
71, 6mpd 15 1 (𝜑𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wex 1703  wnf 1707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-12 2046
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1704  df-nf 1709
This theorem is referenced by:  fvmptd3f  6293  ovmpt2df  6789  ex-natded9.26  27260  exlimimdd  33171  suprnmpt  39177  stoweidlem43  40029  stoweidlem44  40030  stoweidlem54  40040
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