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Theorem r19.29vva 2473
Description: A commonly used pattern based on r19.29 2467, 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 112 . . . . 5 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → (𝜓𝜒))
32ralrimiva 2409 . . . 4 ((𝜑𝑥𝐴) → ∀𝑦𝐵 (𝜓𝜒))
43ralrimiva 2409 . . 3 (𝜑 → ∀𝑥𝐴𝑦𝐵 (𝜓𝜒))
5 r19.29vva.2 . . 3 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)
64, 5r19.29d2r 2472 . 2 (𝜑 → ∃𝑥𝐴𝑦𝐵 ((𝜓𝜒) ∧ 𝜓))
7 pm3.35 333 . . . . 5 ((𝜓 ∧ (𝜓𝜒)) → 𝜒)
87ancoms 259 . . . 4 (((𝜓𝜒) ∧ 𝜓) → 𝜒)
98rexlimivw 2446 . . 3 (∃𝑦𝐵 ((𝜓𝜒) ∧ 𝜓) → 𝜒)
109rexlimivw 2446 . 2 (∃𝑥𝐴𝑦𝐵 ((𝜓𝜒) ∧ 𝜓) → 𝜒)
116, 10syl 14 1 (𝜑𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wcel 1409  wral 2323  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-nf 1366  df-ral 2328  df-rex 2329
This theorem is referenced by: (None)
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