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Theorem r19.29af2 3265
Description: A commonly used pattern based on r19.29 3112. (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 419 . . 3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
62, 3, 5rexlimd 3264 . 2 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
71, 6mpd 15 1 (𝜑𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wnf 1780  wcel 2106  wrex 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  df-ral 3060  df-rex 3069
This theorem is referenced by:  r19.29af  3266  restmetu  24599  opreu2reuALT  32505  aciunf1lem  32679  fprodex01  32832  nsgqusf1olem1  33421  locfinreflem  33801  esumrnmpt2  34049  esum2dlem  34073  esum2d  34074  esumiun  34075
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