Users' Mathboxes Mathbox for Rodolfo Medina < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prtlem13 Structured version   Visualization version   GIF version

Theorem prtlem13 35884
Description: Lemma for prter1 35895, prter2 35897, prter3 35898 and prtex 35896. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
prtlem13.1 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
Assertion
Ref Expression
prtlem13 (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
Distinct variable groups:   𝑣,𝑢,𝑥,𝑦,𝐴   𝑤,𝑣,𝑥,𝑦   𝑧,𝑣,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑧,𝑤)   (𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)

Proof of Theorem prtlem13
StepHypRef Expression
1 vex 3495 . 2 𝑧 ∈ V
2 vex 3495 . 2 𝑤 ∈ V
3 elequ2 2120 . . . . 5 (𝑢 = 𝑣 → (𝑥𝑢𝑥𝑣))
4 elequ2 2120 . . . . 5 (𝑢 = 𝑣 → (𝑦𝑢𝑦𝑣))
53, 4anbi12d 630 . . . 4 (𝑢 = 𝑣 → ((𝑥𝑢𝑦𝑢) ↔ (𝑥𝑣𝑦𝑣)))
65cbvrexvw 3448 . . 3 (∃𝑢𝐴 (𝑥𝑢𝑦𝑢) ↔ ∃𝑣𝐴 (𝑥𝑣𝑦𝑣))
7 elequ1 2112 . . . . 5 (𝑥 = 𝑧 → (𝑥𝑣𝑧𝑣))
8 elequ1 2112 . . . . 5 (𝑦 = 𝑤 → (𝑦𝑣𝑤𝑣))
97, 8bi2anan9 635 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝑣𝑦𝑣) ↔ (𝑧𝑣𝑤𝑣)))
109rexbidv 3294 . . 3 ((𝑥 = 𝑧𝑦 = 𝑤) → (∃𝑣𝐴 (𝑥𝑣𝑦𝑣) ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣)))
116, 10syl5bb 284 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → (∃𝑢𝐴 (𝑥𝑢𝑦𝑢) ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣)))
12 prtlem13.1 . 2 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
131, 2, 11, 12braba 5415 1 (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wrex 3136   class class class wbr 5057  {copab 5119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120
This theorem is referenced by:  prtlem16  35885  prtlem18  35893  prter1  35895  prter3  35898
  Copyright terms: Public domain W3C validator