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Theorem prtlem15 34656
Description: Lemma for prter1 34660 and prtex 34661. (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 887 . . . . . . 7 (((𝑤𝑥𝑤𝑦) ∧ ((𝑢𝑥𝑣𝑦) ∧ (𝑤𝑥𝑤𝑦))) ↔ ((𝑢𝑥𝑣𝑦) ∧ (𝑤𝑥𝑤𝑦)))
2 an43 902 . . . . . . . 8 (((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) ↔ ((𝑢𝑥𝑣𝑦) ∧ (𝑤𝑥𝑤𝑦)))
32anbi2i 732 . . . . . . 7 (((𝑤𝑥𝑤𝑦) ∧ ((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦))) ↔ ((𝑤𝑥𝑤𝑦) ∧ ((𝑢𝑥𝑣𝑦) ∧ (𝑤𝑥𝑤𝑦))))
41, 3, 23bitr4ri 293 . . . . . 6 (((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) ↔ ((𝑤𝑥𝑤𝑦) ∧ ((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦))))
5 prtlem14 34655 . . . . . . . 8 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → ((𝑤𝑥𝑤𝑦) → 𝑥 = 𝑦)))
6 an3 903 . . . . . . . . 9 (((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) → (𝑢𝑥𝑣𝑦))
7 elequ2 2145 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑣𝑥𝑣𝑦))
87anbi2d 742 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑢𝑥𝑣𝑥) ↔ (𝑢𝑥𝑣𝑦)))
96, 8syl5ibr 236 . . . . . . . 8 (𝑥 = 𝑦 → (((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) → (𝑢𝑥𝑣𝑥)))
105, 9syl8 76 . . . . . . 7 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → ((𝑤𝑥𝑤𝑦) → (((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) → (𝑢𝑥𝑣𝑥)))))
1110imp4a 615 . . . . . 6 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → (((𝑤𝑥𝑤𝑦) ∧ ((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦))) → (𝑢𝑥𝑣𝑥))))
124, 11syl7bi 245 . . . . 5 (Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → (((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) → (𝑢𝑥𝑣𝑥))))
1312expdimp 452 . . . 4 ((Prt 𝐴𝑥𝐴) → (𝑦𝐴 → (((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) → (𝑢𝑥𝑣𝑥))))
1413rexlimdv 3160 . . 3 ((Prt 𝐴𝑥𝐴) → (∃𝑦𝐴 ((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) → (𝑢𝑥𝑣𝑥)))
1514reximdva 3147 . 2 (Prt 𝐴 → (∃𝑥𝐴𝑦𝐴 ((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) → ∃𝑥𝐴 (𝑢𝑥𝑣𝑥)))
16 elequ2 2145 . . . 4 (𝑥 = 𝑧 → (𝑢𝑥𝑢𝑧))
17 elequ2 2145 . . . 4 (𝑥 = 𝑧 → (𝑣𝑥𝑣𝑧))
1816, 17anbi12d 749 . . 3 (𝑥 = 𝑧 → ((𝑢𝑥𝑣𝑥) ↔ (𝑢𝑧𝑣𝑧)))
1918cbvrexv 3303 . 2 (∃𝑥𝐴 (𝑢𝑥𝑣𝑥) ↔ ∃𝑧𝐴 (𝑢𝑧𝑣𝑧))
2015, 19syl6ib 241 1 (Prt 𝐴 → (∃𝑥𝐴𝑦𝐴 ((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) → ∃𝑧𝐴 (𝑢𝑧𝑣𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2131  wrex 3043  Prt wprt 34652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-v 3334  df-dif 3710  df-in 3714  df-nul 4051  df-prt 34653
This theorem is referenced by:  prter1  34660
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