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Theorem exlimdd 2213
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 413 . 2 (𝜑 → (𝜓𝜒))
61, 2, 3, 5exlimimdd 2212 1 (𝜑𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  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 1904  ax-6 1963  ax-7 2008  ax-12 2169
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-nf 1778
This theorem is referenced by:  exlimimddOLD  2215  fvmptd3f  6779  ovmpodf  7296  ex-natded9.26  28112  suprnmpt  41295  stoweidlem43  42194  stoweidlem44  42195  stoweidlem54  42205
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