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Theorem prtlem17 37388
Description: Lemma for prter2 37393. (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 3071 . . 3 (∃𝑦𝐴 (𝑧𝑦𝑤𝑦) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧𝑦𝑤𝑦)))
2 an32 645 . . . . . . . 8 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝑥) ↔ ((𝑥𝐴𝑧𝑥) ∧ 𝑦𝐴))
3 prtlem14 37386 . . . . . . . . . . 11 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → ((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)))
4 elequ2 2122 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑤𝑥𝑤𝑦))
54biimprd 248 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑤𝑦𝑤𝑥))
63, 5syl8 76 . . . . . . . . . 10 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → ((𝑧𝑥𝑧𝑦) → (𝑤𝑦𝑤𝑥))))
76exp4a 433 . . . . . . . . 9 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → (𝑧𝑥 → (𝑧𝑦 → (𝑤𝑦𝑤𝑥)))))
87impd 412 . . . . . . . 8 (Prt 𝐴 → (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝑥) → (𝑧𝑦 → (𝑤𝑦𝑤𝑥))))
92, 8biimtrrid 242 . . . . . . 7 (Prt 𝐴 → (((𝑥𝐴𝑧𝑥) ∧ 𝑦𝐴) → (𝑧𝑦 → (𝑤𝑦𝑤𝑥))))
109expd 417 . . . . . 6 (Prt 𝐴 → ((𝑥𝐴𝑧𝑥) → (𝑦𝐴 → (𝑧𝑦 → (𝑤𝑦𝑤𝑥)))))
1110imp5a 442 . . . . 5 (Prt 𝐴 → ((𝑥𝐴𝑧𝑥) → (𝑦𝐴 → ((𝑧𝑦𝑤𝑦) → 𝑤𝑥))))
1211imp4b 423 . . . 4 ((Prt 𝐴 ∧ (𝑥𝐴𝑧𝑥)) → ((𝑦𝐴 ∧ (𝑧𝑦𝑤𝑦)) → 𝑤𝑥))
1312exlimdv 1937 . . 3 ((Prt 𝐴 ∧ (𝑥𝐴𝑧𝑥)) → (∃𝑦(𝑦𝐴 ∧ (𝑧𝑦𝑤𝑦)) → 𝑤𝑥))
141, 13biimtrid 241 . 2 ((Prt 𝐴 ∧ (𝑥𝐴𝑧𝑥)) → (∃𝑦𝐴 (𝑧𝑦𝑤𝑦) → 𝑤𝑥))
1514ex 414 1 (Prt 𝐴 → ((𝑥𝐴𝑧𝑥) → (∃𝑦𝐴 (𝑧𝑦𝑤𝑦) → 𝑤𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wex 1782  wcel 2107  wrex 3070  Prt wprt 37383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-v 3449  df-dif 3917  df-in 3921  df-nul 4287  df-prt 37384
This theorem is referenced by:  prtlem18  37389
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