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Theorem exlimdd 2075
 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 449 . . 3 (𝜑 → (𝜓𝜒))
62, 3, 5exlimd 2074 . 2 (𝜑 → (∃𝑥𝜓𝜒))
71, 6mpd 15 1 (𝜑𝜒)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695  Ⅎwnf 1699 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-12 2034 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701 This theorem is referenced by:  fvmptdf  6204  ovmpt2df  6690  ex-natded9.26  26668  exlimimdd  32367  suprnmpt  38350  stoweidlem43  38936  stoweidlem44  38937  stoweidlem54  38947
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