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Theorem r19.29a 2633
Description: A commonly used pattern based on r19.29 2627. (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 1539 . 2 𝑥𝜑
2 r19.29a.1 . 2 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
3 r19.29a.2 . 2 (𝜑 → ∃𝑥𝐴 𝜓)
41, 2, 3r19.29af 2631 1 (𝜑𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wcel 2160  wrex 2469
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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-ral 2473  df-rex 2474
This theorem is referenced by:  cnegexlem3  8152  cnegex  8153  modqmuladdnn0  10386  uzwodc  12056  1arith  12383  mhmid  13023  mhmmnd  13024  ghmgrp  13026  ghmcmn  13226  ringinvnz1ne0  13362  neitx  14165
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