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Theorem r19.29af2 3267
Description: A commonly used pattern based on r19.29 3114. (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 3266 . 2 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
71, 6mpd 15 1 (𝜑𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wnf 1783  wcel 2108  wrex 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-12 2177
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-ral 3062  df-rex 3071
This theorem is referenced by:  r19.29af  3268  restmetu  24583  opreu2reuALT  32496  aciunf1lem  32672  fprodex01  32827  nsgqusf1olem1  33441  locfinreflem  33839  esumrnmpt2  34069  esum2dlem  34093  esum2d  34094  esumiun  34095
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