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Theorem r19.29a 2471
 Description: A commonly used pattern based on r19.29 2467 (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 1437 . 2 𝑥𝜑
2 r19.29a.1 . 2 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
3 r19.29a.2 . 2 (𝜑 → ∃𝑥𝐴 𝜓)
41, 2, 3r19.29af 2470 1 (𝜑𝜒)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ∈ wcel 1409  ∃wrex 2324 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-i5r 1444 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-ral 2328  df-rex 2329 This theorem is referenced by:  cnegexlem3  7251  cnegex  7252  modqmuladdnn0  9318
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