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Theorem exlimdd 1882
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 115 . . 3 (𝜑 → (𝜓𝜒))
62, 3, 5exlimd 1607 . 2 (𝜑 → (∃𝑥𝜓𝜒))
71, 6mpd 13 1 (𝜑𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wnf 1470  wex 1502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia3 108  ax-5 1457  ax-gen 1459  ax-ie2 1504  ax-4 1520
This theorem depends on definitions:  df-bi 117  df-nf 1471
This theorem is referenced by:  fvmptdf  5616  ovmpodf  6019  exmidfodomrlemr  7214  exmidfodomrlemrALT  7215  ltexprlemm  7612  dfgrp3mlem  12992
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