Mathbox for Rodolfo Medina < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prtlem15 Structured version   Visualization version   GIF version

Theorem prtlem15 36190
 Description: Lemma for prter1 36194 and prtex 36195. (Contributed by Rodolfo Medina, 13-Oct-2010.)
Assertion
Ref Expression
prtlem15 (Prt 𝐴 → (∃𝑥𝐴𝑦𝐴 ((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) → ∃𝑧𝐴 (𝑢𝑧𝑣𝑧)))
Distinct variable groups:   𝑣,𝑢,𝑤,𝑥,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑤,𝑣,𝑢)

Proof of Theorem prtlem15
StepHypRef Expression
1 anabs7 663 . . . . . . 7 (((𝑤𝑥𝑤𝑦) ∧ ((𝑢𝑥𝑣𝑦) ∧ (𝑤𝑥𝑤𝑦))) ↔ ((𝑢𝑥𝑣𝑦) ∧ (𝑤𝑥𝑤𝑦)))
2 an43 657 . . . . . . . 8 (((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) ↔ ((𝑢𝑥𝑣𝑦) ∧ (𝑤𝑥𝑤𝑦)))
32anbi2i 625 . . . . . . 7 (((𝑤𝑥𝑤𝑦) ∧ ((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦))) ↔ ((𝑤𝑥𝑤𝑦) ∧ ((𝑢𝑥𝑣𝑦) ∧ (𝑤𝑥𝑤𝑦))))
41, 3, 23bitr4ri 307 . . . . . 6 (((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) ↔ ((𝑤𝑥𝑤𝑦) ∧ ((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦))))
5 prtlem14 36189 . . . . . . . 8 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → ((𝑤𝑥𝑤𝑦) → 𝑥 = 𝑦)))
6 an3 658 . . . . . . . . 9 (((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) → (𝑢𝑥𝑣𝑦))
7 elequ2 2126 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑣𝑥𝑣𝑦))
87anbi2d 631 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑢𝑥𝑣𝑥) ↔ (𝑢𝑥𝑣𝑦)))
96, 8syl5ibr 249 . . . . . . . 8 (𝑥 = 𝑦 → (((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) → (𝑢𝑥𝑣𝑥)))
105, 9syl8 76 . . . . . . 7 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → ((𝑤𝑥𝑤𝑦) → (((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) → (𝑢𝑥𝑣𝑥)))))
1110imp4a 426 . . . . . 6 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → (((𝑤𝑥𝑤𝑦) ∧ ((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦))) → (𝑢𝑥𝑣𝑥))))
124, 11syl7bi 258 . . . . 5 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → (((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) → (𝑢𝑥𝑣𝑥))))
1312expdimp 456 . . . 4 ((Prt 𝐴𝑥𝐴) → (𝑦𝐴 → (((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) → (𝑢𝑥𝑣𝑥))))
1413rexlimdv 3242 . . 3 ((Prt 𝐴𝑥𝐴) → (∃𝑦𝐴 ((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) → (𝑢𝑥𝑣𝑥)))
1514reximdva 3233 . 2 (Prt 𝐴 → (∃𝑥𝐴𝑦𝐴 ((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) → ∃𝑥𝐴 (𝑢𝑥𝑣𝑥)))
16 elequ2 2126 . . . 4 (𝑥 = 𝑧 → (𝑢𝑥𝑢𝑧))
17 elequ2 2126 . . . 4 (𝑥 = 𝑧 → (𝑣𝑥𝑣𝑧))
1816, 17anbi12d 633 . . 3 (𝑥 = 𝑧 → ((𝑢𝑥𝑣𝑥) ↔ (𝑢𝑧𝑣𝑧)))
1918cbvrexvw 3397 . 2 (∃𝑥𝐴 (𝑢𝑥𝑣𝑥) ↔ ∃𝑧𝐴 (𝑢𝑧𝑣𝑧))
2015, 19syl6ib 254 1 (Prt 𝐴 → (∃𝑥𝐴𝑦𝐴 ((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) → ∃𝑧𝐴 (𝑢𝑧𝑣𝑧)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  ∃wrex 3107  Prt wprt 36186 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 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-in 3888  df-nul 4244  df-prt 36187 This theorem is referenced by:  prter1  36194
 Copyright terms: Public domain W3C validator