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Theorem exlimdd 1844
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 114 . . 3 (𝜑 → (𝜓𝜒))
62, 3, 5exlimd 1576 . 2 (𝜑 → (∃𝑥𝜓𝜒))
71, 6mpd 13 1 (𝜑𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wnf 1436  wex 1468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie2 1470  ax-4 1487
This theorem depends on definitions:  df-bi 116  df-nf 1437
This theorem is referenced by:  fvmptdf  5508  ovmpodf  5902  exmidfodomrlemr  7058  exmidfodomrlemrALT  7059  ltexprlemm  7415
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