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Theorem prtlem18 33681
Description: Lemma for prter2 33685. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
prtlem18.1 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
Assertion
Ref Expression
prtlem18 (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → (𝑤𝑣𝑧 𝑤)))
Distinct variable groups:   𝑣,𝑢,𝑤,𝑥,𝑦,𝑧,𝐴   𝑣, ,𝑤,𝑧
Allowed substitution hints:   (𝑥,𝑦,𝑢)

Proof of Theorem prtlem18
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 rspe 2999 . . . . 5 ((𝑣𝐴 ∧ (𝑧𝑣𝑤𝑣)) → ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
21expr 642 . . . 4 ((𝑣𝐴𝑧𝑣) → (𝑤𝑣 → ∃𝑣𝐴 (𝑧𝑣𝑤𝑣)))
3 prtlem18.1 . . . . 5 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
43prtlem13 33672 . . . 4 (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
52, 4syl6ibr 242 . . 3 ((𝑣𝐴𝑧𝑣) → (𝑤𝑣𝑧 𝑤))
65a1i 11 . 2 (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → (𝑤𝑣𝑧 𝑤)))
73prtlem13 33672 . . 3 (𝑧 𝑤 ↔ ∃𝑝𝐴 (𝑧𝑝𝑤𝑝))
8 prtlem17 33680 . . 3 (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → (∃𝑝𝐴 (𝑧𝑝𝑤𝑝) → 𝑤𝑣)))
97, 8syl7bi 245 . 2 (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → (𝑧 𝑤𝑤𝑣)))
106, 9impbidd 200 1 (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → (𝑤𝑣𝑧 𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2909   class class class wbr 4623  {copab 4682  Prt wprt 33675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-prt 33676
This theorem is referenced by:  prtlem19  33682
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