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Theorem prtlem18 36131
Description: Lemma for prter2 36135. (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 3290 . . . . 5 ((𝑣𝐴 ∧ (𝑧𝑣𝑤𝑣)) → ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
21expr 460 . . . 4 ((𝑣𝐴𝑧𝑣) → (𝑤𝑣 → ∃𝑣𝐴 (𝑧𝑣𝑤𝑣)))
3 prtlem18.1 . . . . 5 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
43prtlem13 36122 . . . 4 (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
52, 4syl6ibr 255 . . 3 ((𝑣𝐴𝑧𝑣) → (𝑤𝑣𝑧 𝑤))
65a1i 11 . 2 (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → (𝑤𝑣𝑧 𝑤)))
73prtlem13 36122 . . 3 (𝑧 𝑤 ↔ ∃𝑝𝐴 (𝑧𝑝𝑤𝑝))
8 prtlem17 36130 . . 3 (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → (∃𝑝𝐴 (𝑧𝑝𝑤𝑝) → 𝑤𝑣)))
97, 8syl7bi 258 . 2 (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → (𝑧 𝑤𝑤𝑣)))
106, 9impbidd 213 1 (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → (𝑤𝑣𝑧 𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  wrex 3131   class class class wbr 5042  {copab 5104  Prt wprt 36125
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-prt 36126
This theorem is referenced by:  prtlem19  36132
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