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Theorem exlimdd 2220
Description: Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 3-Sep-2023.)
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.1 . 2 𝑥𝜑
2 exlimdd.2 . 2 𝑥𝜒
3 exlimdd.3 . 2 (𝜑 → ∃𝑥𝜓)
4 exlimdd.4 . . 3 ((𝜑𝜓) → 𝜒)
54ex 416 . 2 (𝜑 → (𝜓𝜒))
61, 2, 3, 5exlimimdd 2219 1 (𝜑𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wex 1787  wnf 1791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-12 2177
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-nf 1792
This theorem is referenced by:  fvmptd3f  6855  ovmpodf  7387  ex-natded9.26  28534  stoweidlem43  43305  stoweidlem44  43306  stoweidlem54  43316
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