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Theorem exlimimdd 2212
Description: Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.) Shorten exlimdd 2213. (Revised by Wolf Lammen, 3-Sep-2023.)
Hypotheses
Ref Expression
exlimdd.1 𝑥𝜑
exlimdd.2 𝑥𝜒
exlimdd.3 (𝜑 → ∃𝑥𝜓)
exlimimdd.4 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
exlimimdd (𝜑𝜒)

Proof of Theorem exlimimdd
StepHypRef Expression
1 exlimdd.3 . 2 (𝜑 → ∃𝑥𝜓)
2 exlimdd.1 . . 3 𝑥𝜑
3 exlimdd.2 . . 3 𝑥𝜒
4 exlimimdd.4 . . 3 (𝜑 → (𝜓𝜒))
52, 3, 4exlimd 2211 . 2 (𝜑 → (∃𝑥𝜓𝜒))
61, 5mpd 15 1 (𝜑𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1774  wnf 1778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-12 2170
This theorem depends on definitions:  df-bi 209  df-ex 1775  df-nf 1779
This theorem is referenced by:  exlimdd  2213  tz6.12c  6688  ovmpodf  7298  gsum2d2lem  19085  padct  30447  stoweidlem27  42303  intsaluni  42603  isomenndlem  42803  tz6.12c-afv2  43432
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