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Theorem prtlem17 34665
Description: Lemma for prter2 34670. (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 3056 . . 3 (∃𝑦𝐴 (𝑧𝑦𝑤𝑦) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧𝑦𝑤𝑦)))
2 an32 874 . . . . . . . 8 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝑥) ↔ ((𝑥𝐴𝑧𝑥) ∧ 𝑦𝐴))
3 prtlem14 34663 . . . . . . . . . . 11 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → ((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)))
4 elequ2 2153 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑤𝑥𝑤𝑦))
54biimprd 238 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑤𝑦𝑤𝑥))
63, 5syl8 76 . . . . . . . . . 10 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → ((𝑧𝑥𝑧𝑦) → (𝑤𝑦𝑤𝑥))))
76exp4a 634 . . . . . . . . 9 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → (𝑧𝑥 → (𝑧𝑦 → (𝑤𝑦𝑤𝑥)))))
87impd 446 . . . . . . . 8 (Prt 𝐴 → (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝑥) → (𝑧𝑦 → (𝑤𝑦𝑤𝑥))))
92, 8syl5bir 233 . . . . . . 7 (Prt 𝐴 → (((𝑥𝐴𝑧𝑥) ∧ 𝑦𝐴) → (𝑧𝑦 → (𝑤𝑦𝑤𝑥))))
109expd 451 . . . . . 6 (Prt 𝐴 → ((𝑥𝐴𝑧𝑥) → (𝑦𝐴 → (𝑧𝑦 → (𝑤𝑦𝑤𝑥)))))
1110imp5a 625 . . . . 5 (Prt 𝐴 → ((𝑥𝐴𝑧𝑥) → (𝑦𝐴 → ((𝑧𝑦𝑤𝑦) → 𝑤𝑥))))
1211imp4b 614 . . . 4 ((Prt 𝐴 ∧ (𝑥𝐴𝑧𝑥)) → ((𝑦𝐴 ∧ (𝑧𝑦𝑤𝑦)) → 𝑤𝑥))
1312exlimdv 2010 . . 3 ((Prt 𝐴 ∧ (𝑥𝐴𝑧𝑥)) → (∃𝑦(𝑦𝐴 ∧ (𝑧𝑦𝑤𝑦)) → 𝑤𝑥))
141, 13syl5bi 232 . 2 ((Prt 𝐴 ∧ (𝑥𝐴𝑧𝑥)) → (∃𝑦𝐴 (𝑧𝑦𝑤𝑦) → 𝑤𝑥))
1514ex 449 1 (Prt 𝐴 → ((𝑥𝐴𝑧𝑥) → (∃𝑦𝐴 (𝑧𝑦𝑤𝑦) → 𝑤𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wex 1853  wcel 2139  wrex 3051  Prt wprt 34660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-dif 3718  df-in 3722  df-nul 4059  df-prt 34661
This theorem is referenced by:  prtlem18  34666
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