Users' Mathboxes Mathbox for Rodolfo Medina < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prtlem17 Structured version   Visualization version   GIF version

Theorem prtlem17 38210
Description: Lemma for prter2 38215. (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 3070 . . 3 (∃𝑦𝐴 (𝑧𝑦𝑤𝑦) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧𝑦𝑤𝑦)))
2 an32 643 . . . . . . . 8 (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝑥) ↔ ((𝑥𝐴𝑧𝑥) ∧ 𝑦𝐴))
3 prtlem14 38208 . . . . . . . . . . 11 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → ((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)))
4 elequ2 2120 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑤𝑥𝑤𝑦))
54biimprd 247 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑤𝑦𝑤𝑥))
63, 5syl8 76 . . . . . . . . . 10 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → ((𝑧𝑥𝑧𝑦) → (𝑤𝑦𝑤𝑥))))
76exp4a 431 . . . . . . . . 9 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → (𝑧𝑥 → (𝑧𝑦 → (𝑤𝑦𝑤𝑥)))))
87impd 410 . . . . . . . 8 (Prt 𝐴 → (((𝑥𝐴𝑦𝐴) ∧ 𝑧𝑥) → (𝑧𝑦 → (𝑤𝑦𝑤𝑥))))
92, 8biimtrrid 242 . . . . . . 7 (Prt 𝐴 → (((𝑥𝐴𝑧𝑥) ∧ 𝑦𝐴) → (𝑧𝑦 → (𝑤𝑦𝑤𝑥))))
109expd 415 . . . . . 6 (Prt 𝐴 → ((𝑥𝐴𝑧𝑥) → (𝑦𝐴 → (𝑧𝑦 → (𝑤𝑦𝑤𝑥)))))
1110imp5a 440 . . . . 5 (Prt 𝐴 → ((𝑥𝐴𝑧𝑥) → (𝑦𝐴 → ((𝑧𝑦𝑤𝑦) → 𝑤𝑥))))
1211imp4b 421 . . . 4 ((Prt 𝐴 ∧ (𝑥𝐴𝑧𝑥)) → ((𝑦𝐴 ∧ (𝑧𝑦𝑤𝑦)) → 𝑤𝑥))
1312exlimdv 1935 . . 3 ((Prt 𝐴 ∧ (𝑥𝐴𝑧𝑥)) → (∃𝑦(𝑦𝐴 ∧ (𝑧𝑦𝑤𝑦)) → 𝑤𝑥))
141, 13biimtrid 241 . 2 ((Prt 𝐴 ∧ (𝑥𝐴𝑧𝑥)) → (∃𝑦𝐴 (𝑧𝑦𝑤𝑦) → 𝑤𝑥))
1514ex 412 1 (Prt 𝐴 → ((𝑥𝐴𝑧𝑥) → (∃𝑦𝐴 (𝑧𝑦𝑤𝑦) → 𝑤𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1780  wcel 2105  wrex 3069  Prt wprt 38205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-v 3475  df-dif 3951  df-in 3955  df-nul 4323  df-prt 38206
This theorem is referenced by:  prtlem18  38211
  Copyright terms: Public domain W3C validator