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Theorem r19.29af 2628
Description: A commonly used pattern based on r19.29 2624. (Contributed by Thierry Arnoux, 29-Nov-2017.)
Hypotheses
Ref Expression
r19.29af.0 𝑥𝜑
r19.29af.1 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
r19.29af.2 (𝜑 → ∃𝑥𝐴 𝜓)
Assertion
Ref Expression
r19.29af (𝜑𝜒)
Distinct variable group:   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.29af
StepHypRef Expression
1 r19.29af.0 . 2 𝑥𝜑
2 nfv 1538 . 2 𝑥𝜒
3 r19.29af.1 . 2 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
4 r19.29af.2 . 2 (𝜑 → ∃𝑥𝐴 𝜓)
51, 2, 3, 4r19.29af2 2627 1 (𝜑𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wnf 1470  wcel 2158  wrex 2466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-4 1520  ax-17 1536  ax-ial 1544  ax-i5r 1545
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-ral 2470  df-rex 2471
This theorem is referenced by:  r19.29an  2629  r19.29a  2630  suplocsrlem  7820  supinfneg  9608  infsupneg  9609  pw1nct  14993
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