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Theorem r19.29a 2511
Description: A commonly used pattern based on r19.29 2506 (Contributed by Thierry Arnoux, 22-Nov-2017.)
Hypotheses
Ref Expression
r19.29a.1 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
r19.29a.2 (𝜑 → ∃𝑥𝐴 𝜓)
Assertion
Ref Expression
r19.29a (𝜑𝜒)
Distinct variable groups:   𝜒,𝑥   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.29a
StepHypRef Expression
1 nfv 1466 . 2 𝑥𝜑
2 r19.29a.1 . 2 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
3 r19.29a.2 . 2 (𝜑 → ∃𝑥𝐴 𝜓)
41, 2, 3r19.29af 2509 1 (𝜑𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wcel 1438  wrex 2360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-ral 2364  df-rex 2365
This theorem is referenced by:  cnegexlem3  7659  cnegex  7660  modqmuladdnn0  9775
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