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Theorem prtlem17 35564
Description: Lemma for prter2 35569. (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 3113 . . 3 (∃𝑦𝐴 (𝑧𝑦𝑤𝑦) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧𝑦𝑤𝑦)))
2 an32 642 . . . . . . . 8 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝑥) ↔ ((𝑥𝐴𝑧𝑥) ∧ 𝑦𝐴))
3 prtlem14 35562 . . . . . . . . . . 11 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → ((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)))
4 elequ2 2098 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑤𝑥𝑤𝑦))
54biimprd 249 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑤𝑦𝑤𝑥))
63, 5syl8 76 . . . . . . . . . 10 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → ((𝑧𝑥𝑧𝑦) → (𝑤𝑦𝑤𝑥))))
76exp4a 432 . . . . . . . . 9 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → (𝑧𝑥 → (𝑧𝑦 → (𝑤𝑦𝑤𝑥)))))
87impd 411 . . . . . . . 8 (Prt 𝐴 → (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝑥) → (𝑧𝑦 → (𝑤𝑦𝑤𝑥))))
92, 8syl5bir 244 . . . . . . 7 (Prt 𝐴 → (((𝑥𝐴𝑧𝑥) ∧ 𝑦𝐴) → (𝑧𝑦 → (𝑤𝑦𝑤𝑥))))
109expd 416 . . . . . 6 (Prt 𝐴 → ((𝑥𝐴𝑧𝑥) → (𝑦𝐴 → (𝑧𝑦 → (𝑤𝑦𝑤𝑥)))))
1110imp5a 441 . . . . 5 (Prt 𝐴 → ((𝑥𝐴𝑧𝑥) → (𝑦𝐴 → ((𝑧𝑦𝑤𝑦) → 𝑤𝑥))))
1211imp4b 422 . . . 4 ((Prt 𝐴 ∧ (𝑥𝐴𝑧𝑥)) → ((𝑦𝐴 ∧ (𝑧𝑦𝑤𝑦)) → 𝑤𝑥))
1312exlimdv 1915 . . 3 ((Prt 𝐴 ∧ (𝑥𝐴𝑧𝑥)) → (∃𝑦(𝑦𝐴 ∧ (𝑧𝑦𝑤𝑦)) → 𝑤𝑥))
141, 13syl5bi 243 . 2 ((Prt 𝐴 ∧ (𝑥𝐴𝑧𝑥)) → (∃𝑦𝐴 (𝑧𝑦𝑤𝑦) → 𝑤𝑥))
1514ex 413 1 (Prt 𝐴 → ((𝑥𝐴𝑧𝑥) → (∃𝑦𝐴 (𝑧𝑦𝑤𝑦) → 𝑤𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wex 1765  wcel 2083  wrex 3108  Prt wprt 35559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-v 3442  df-dif 3868  df-in 3872  df-nul 4218  df-prt 35560
This theorem is referenced by:  prtlem18  35565
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