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Theorem r19.29af2 3321
 Description: A commonly used pattern based on r19.29 3248. (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 423 . . 3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
62, 3, 5rexlimd 3309 . 2 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
71, 6mpd 15 1 (𝜑𝜒)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  Ⅎwnf 1785   ∈ wcel 2115  ∃wrex 3133 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-12 2179 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-ral 3137  df-rex 3138 This theorem is referenced by:  r19.29af  3322  restmetu  23168  opreu2reuALT  30237  aciunf1lem  30406  fprodex01  30540  locfinreflem  31127  esumrnmpt2  31347  esum2dlem  31371  esum2d  31372  esumiun  31373
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