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Theorem prtlem18 34685
Description: Lemma for prter2 34689. (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 3151 . . . . 5 ((𝑣𝐴 ∧ (𝑧𝑣𝑤𝑣)) → ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
21expr 444 . . . 4 ((𝑣𝐴𝑧𝑣) → (𝑤𝑣 → ∃𝑣𝐴 (𝑧𝑣𝑤𝑣)))
3 prtlem18.1 . . . . 5 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
43prtlem13 34676 . . . 4 (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
52, 4syl6ibr 242 . . 3 ((𝑣𝐴𝑧𝑣) → (𝑤𝑣𝑧 𝑤))
65a1i 11 . 2 (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → (𝑤𝑣𝑧 𝑤)))
73prtlem13 34676 . . 3 (𝑧 𝑤 ↔ ∃𝑝𝐴 (𝑧𝑝𝑤𝑝))
8 prtlem17 34684 . . 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 382   = wceq 1631  wcel 2145  wrex 3062   class class class wbr 4786  {copab 4846  Prt wprt 34679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-prt 34680
This theorem is referenced by:  prtlem19  34686
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