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Theorem prtlem13 36479
 Description: Lemma for prter1 36490, prter2 36492, prter3 36493 and prtex 36491. (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 3413 . 2 𝑧 ∈ V
2 vex 3413 . 2 𝑤 ∈ V
3 elequ2 2126 . . . . 5 (𝑢 = 𝑣 → (𝑥𝑢𝑥𝑣))
4 elequ2 2126 . . . . 5 (𝑢 = 𝑣 → (𝑦𝑢𝑦𝑣))
53, 4anbi12d 633 . . . 4 (𝑢 = 𝑣 → ((𝑥𝑢𝑦𝑢) ↔ (𝑥𝑣𝑦𝑣)))
65cbvrexvw 3362 . . 3 (∃𝑢𝐴 (𝑥𝑢𝑦𝑢) ↔ ∃𝑣𝐴 (𝑥𝑣𝑦𝑣))
7 elequ1 2118 . . . . 5 (𝑥 = 𝑧 → (𝑥𝑣𝑧𝑣))
8 elequ1 2118 . . . . 5 (𝑦 = 𝑤 → (𝑦𝑣𝑤𝑣))
97, 8bi2anan9 638 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝑣𝑦𝑣) ↔ (𝑧𝑣𝑤𝑣)))
109rexbidv 3221 . . 3 ((𝑥 = 𝑧𝑦 = 𝑤) → (∃𝑣𝐴 (𝑥𝑣𝑦𝑣) ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣)))
116, 10syl5bb 286 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → (∃𝑢𝐴 (𝑥𝑢𝑦𝑢) ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣)))
12 prtlem13.1 . 2 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
131, 2, 11, 12braba 5398 1 (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wrex 3071   class class class wbr 5036  {copab 5098 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 2113  ax-9 2121  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-rex 3076  df-v 3411  df-dif 3863  df-un 3865  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-opab 5099 This theorem is referenced by:  prtlem16  36480  prtlem18  36488  prter1  36490  prter3  36493
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