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Theorem r19.29an 2572
 Description: A commonly used pattern based on r19.29 2567. (Contributed by Thierry Arnoux, 29-Dec-2019.)
Hypothesis
Ref Expression
r19.29an.1 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
Assertion
Ref Expression
r19.29an ((𝜑 ∧ ∃𝑥𝐴 𝜓) → 𝜒)
Distinct variable groups:   𝜒,𝑥   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.29an
StepHypRef Expression
1 nfv 1508 . . 3 𝑥𝜑
2 nfre1 2474 . . 3 𝑥𝑥𝐴 𝜓
31, 2nfan 1544 . 2 𝑥(𝜑 ∧ ∃𝑥𝐴 𝜓)
4 r19.29an.1 . . 3 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
54adantllr 472 . 2 ((((𝜑 ∧ ∃𝑥𝐴 𝜓) ∧ 𝑥𝐴) ∧ 𝜓) → 𝜒)
6 simpr 109 . 2 ((𝜑 ∧ ∃𝑥𝐴 𝜓) → ∃𝑥𝐴 𝜓)
73, 5, 6r19.29af 2571 1 ((𝜑 ∧ ∃𝑥𝐴 𝜓) → 𝜒)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 1480  ∃wrex 2415 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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-i5r 1515 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-ral 2419  df-rex 2420 This theorem is referenced by:  summodclem2  11144
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