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Theorem r19.29af2 2597
 Description: A commonly used pattern based on r19.29 2594 (Contributed by Thierry Arnoux, 17-Dec-2017.)
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 . . 3 (𝜑 → ∃𝑥𝐴 𝜓)
2 r19.29af2.p . . . 4 𝑥𝜑
3 r19.29af2.1 . . . . 5 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
43exp31 362 . . . 4 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
52, 4ralrimi 2528 . . 3 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
61, 5jca 304 . 2 (𝜑 → (∃𝑥𝐴 𝜓 ∧ ∀𝑥𝐴 (𝜓𝜒)))
7 r19.29r 2595 . 2 ((∃𝑥𝐴 𝜓 ∧ ∀𝑥𝐴 (𝜓𝜒)) → ∃𝑥𝐴 (𝜓 ∧ (𝜓𝜒)))
8 r19.29af2.c . . 3 𝑥𝜒
9 pm3.35 345 . . . 4 ((𝜓 ∧ (𝜓𝜒)) → 𝜒)
109a1i 9 . . 3 (𝑥𝐴 → ((𝜓 ∧ (𝜓𝜒)) → 𝜒))
118, 10rexlimi 2567 . 2 (∃𝑥𝐴 (𝜓 ∧ (𝜓𝜒)) → 𝜒)
126, 7, 113syl 17 1 (𝜑𝜒)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  Ⅎwnf 1440   ∈ wcel 2128  ∀wral 2435  ∃wrex 2436 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514  ax-i5r 1515 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-ral 2440  df-rex 2441 This theorem is referenced by:  r19.29af  2598  ctiunctlemfo  12140
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