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Theorem prtlem17 36487
 Description: Lemma for prter2 36492. (Contributed by Rodolfo Medina, 15-Oct-2010.)
Assertion
Ref Expression
prtlem17 (Prt 𝐴 → ((𝑥𝐴𝑧𝑥) → (∃𝑦𝐴 (𝑧𝑦𝑤𝑦) → 𝑤𝑥)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑧,𝑦   𝑦,𝑤
Allowed substitution hints:   𝐴(𝑧,𝑤)

Proof of Theorem prtlem17
StepHypRef Expression
1 df-rex 3076 . . 3 (∃𝑦𝐴 (𝑧𝑦𝑤𝑦) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧𝑦𝑤𝑦)))
2 an32 645 . . . . . . . 8 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝑥) ↔ ((𝑥𝐴𝑧𝑥) ∧ 𝑦𝐴))
3 prtlem14 36485 . . . . . . . . . . 11 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → ((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)))
4 elequ2 2126 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑤𝑥𝑤𝑦))
54biimprd 251 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑤𝑦𝑤𝑥))
63, 5syl8 76 . . . . . . . . . 10 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → ((𝑧𝑥𝑧𝑦) → (𝑤𝑦𝑤𝑥))))
76exp4a 435 . . . . . . . . 9 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → (𝑧𝑥 → (𝑧𝑦 → (𝑤𝑦𝑤𝑥)))))
87impd 414 . . . . . . . 8 (Prt 𝐴 → (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝑥) → (𝑧𝑦 → (𝑤𝑦𝑤𝑥))))
92, 8syl5bir 246 . . . . . . 7 (Prt 𝐴 → (((𝑥𝐴𝑧𝑥) ∧ 𝑦𝐴) → (𝑧𝑦 → (𝑤𝑦𝑤𝑥))))
109expd 419 . . . . . 6 (Prt 𝐴 → ((𝑥𝐴𝑧𝑥) → (𝑦𝐴 → (𝑧𝑦 → (𝑤𝑦𝑤𝑥)))))
1110imp5a 444 . . . . 5 (Prt 𝐴 → ((𝑥𝐴𝑧𝑥) → (𝑦𝐴 → ((𝑧𝑦𝑤𝑦) → 𝑤𝑥))))
1211imp4b 425 . . . 4 ((Prt 𝐴 ∧ (𝑥𝐴𝑧𝑥)) → ((𝑦𝐴 ∧ (𝑧𝑦𝑤𝑦)) → 𝑤𝑥))
1312exlimdv 1934 . . 3 ((Prt 𝐴 ∧ (𝑥𝐴𝑧𝑥)) → (∃𝑦(𝑦𝐴 ∧ (𝑧𝑦𝑤𝑦)) → 𝑤𝑥))
141, 13syl5bi 245 . 2 ((Prt 𝐴 ∧ (𝑥𝐴𝑧𝑥)) → (∃𝑦𝐴 (𝑧𝑦𝑤𝑦) → 𝑤𝑥))
1514ex 416 1 (Prt 𝐴 → ((𝑥𝐴𝑧𝑥) → (∃𝑦𝐴 (𝑧𝑦𝑤𝑦) → 𝑤𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1781   ∈ wcel 2111  ∃wrex 3071  Prt wprt 36482 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3863  df-in 3867  df-nul 4228  df-prt 36483 This theorem is referenced by:  prtlem18  36488
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