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Theorem r19.29af2 3057
 Description: A commonly used pattern based on r19.29 3054. (Contributed by Thierry Arnoux, 17-Dec-2017.) (Proof shortened by OpenAI, 25-Mar-2020.)
Hypotheses
Ref Expression
r19.29af2.p 𝑥𝜑
r19.29af2.c 𝑥𝜒
r19.29af2.1 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
r19.29af2.2 (𝜑 → ∃𝑥𝐴 𝜓)
Assertion
Ref Expression
r19.29af2 (𝜑𝜒)

Proof of Theorem r19.29af2
StepHypRef Expression
1 r19.29af2.2 . 2 (𝜑 → ∃𝑥𝐴 𝜓)
2 r19.29af2.p . . 3 𝑥𝜑
3 r19.29af2.c . . 3 𝑥𝜒
4 r19.29af2.1 . . . 4 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
54exp31 628 . . 3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
62, 3, 5rexlimd 3008 . 2 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
71, 6mpd 15 1 (𝜑𝜒)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  Ⅎwnf 1699   ∈ wcel 1977  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701  df-ral 2901  df-rex 2902 This theorem is referenced by:  r19.29af  3058  restmetu  22185  aciunf1lem  28844  locfinreflem  29235  esumrnmpt2  29457  esum2dlem  29481  esum2d  29482  esumiun  29483
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