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Theorem r19.29vva 2577
 Description: A commonly used pattern based on r19.29 2569, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)
Hypotheses
Ref Expression
r19.29vva.1 ((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)
r19.29vva.2 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)
Assertion
Ref Expression
r19.29vva (𝜑𝜒)
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦,𝜒   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem r19.29vva
StepHypRef Expression
1 r19.29vva.1 . . . . . 6 ((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)
21ex 114 . . . . 5 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → (𝜓𝜒))
32ralrimiva 2505 . . . 4 ((𝜑𝑥𝐴) → ∀𝑦𝐵 (𝜓𝜒))
43ralrimiva 2505 . . 3 (𝜑 → ∀𝑥𝐴𝑦𝐵 (𝜓𝜒))
5 r19.29vva.2 . . 3 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)
64, 5r19.29d2r 2576 . 2 (𝜑 → ∃𝑥𝐴𝑦𝐵 ((𝜓𝜒) ∧ 𝜓))
7 pm3.35 344 . . . . 5 ((𝜓 ∧ (𝜓𝜒)) → 𝜒)
87ancoms 266 . . . 4 (((𝜓𝜒) ∧ 𝜓) → 𝜒)
98rexlimivw 2545 . . 3 (∃𝑦𝐵 ((𝜓𝜒) ∧ 𝜓) → 𝜒)
109rexlimivw 2545 . 2 (∃𝑥𝐴𝑦𝐵 ((𝜓𝜒) ∧ 𝜓) → 𝜒)
116, 10syl 14 1 (𝜑𝜒)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417 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-nf 1437  df-ral 2421  df-rex 2422 This theorem is referenced by: (None)
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