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Theorem exlimimdd 2204
Description: Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.) Shorten exlimdd 2205. (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 2203 . 2 (𝜑 → (∃𝑥𝜓𝜒))
61, 5mpd 15 1 (𝜑𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1773  wnf 1777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-ex 1774  df-nf 1778
This theorem is referenced by:  exlimdd  2205  tz6.12cOLD  6912  ovmpodf  7560  gsum2d2lem  19893  2ndresdju  32383  padct  32451  stoweidlem27  45315  intsaluni  45617  isomenndlem  45818  tz6.12c-afv2  46522
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