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Theorem prtlem13 33171
 Description: Lemma for prter1 33182, prter2 33184, prter3 33185 and prtex 33183. (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 3176 . 2 𝑧 ∈ V
2 vex 3176 . 2 𝑤 ∈ V
3 elequ2 1991 . . . . 5 (𝑢 = 𝑣 → (𝑥𝑢𝑥𝑣))
4 elequ2 1991 . . . . 5 (𝑢 = 𝑣 → (𝑦𝑢𝑦𝑣))
53, 4anbi12d 743 . . . 4 (𝑢 = 𝑣 → ((𝑥𝑢𝑦𝑢) ↔ (𝑥𝑣𝑦𝑣)))
65cbvrexv 3148 . . 3 (∃𝑢𝐴 (𝑥𝑢𝑦𝑢) ↔ ∃𝑣𝐴 (𝑥𝑣𝑦𝑣))
7 eleq1 2676 . . . . 5 (𝑥 = 𝑧 → (𝑥𝑣𝑧𝑣))
8 eleq1 2676 . . . . 5 (𝑦 = 𝑤 → (𝑦𝑣𝑤𝑣))
97, 8bi2anan9 913 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝑣𝑦𝑣) ↔ (𝑧𝑣𝑤𝑣)))
109rexbidv 3034 . . 3 ((𝑥 = 𝑧𝑦 = 𝑤) → (∃𝑣𝐴 (𝑥𝑣𝑦𝑣) ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣)))
116, 10syl5bb 271 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → (∃𝑢𝐴 (𝑥𝑢𝑦𝑢) ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣)))
12 prtlem13.1 . 2 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
131, 2, 11, 12braba 4917 1 (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wrex 2897   class class class wbr 4583  {copab 4642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644 This theorem is referenced by:  prtlem16  33172  prtlem18  33180  prter1  33182  prter3  33185
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