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Theorem prtlem13 38251
Description: Lemma for prter1 38262, prter2 38264, prter3 38265 and prtex 38263. (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 3472 . 2 𝑧 ∈ V
2 vex 3472 . 2 𝑤 ∈ V
3 elequ2 2113 . . . . 5 (𝑢 = 𝑣 → (𝑥𝑢𝑥𝑣))
4 elequ2 2113 . . . . 5 (𝑢 = 𝑣 → (𝑦𝑢𝑦𝑣))
53, 4anbi12d 630 . . . 4 (𝑢 = 𝑣 → ((𝑥𝑢𝑦𝑢) ↔ (𝑥𝑣𝑦𝑣)))
65cbvrexvw 3229 . . 3 (∃𝑢𝐴 (𝑥𝑢𝑦𝑢) ↔ ∃𝑣𝐴 (𝑥𝑣𝑦𝑣))
7 elequ1 2105 . . . . 5 (𝑥 = 𝑧 → (𝑥𝑣𝑧𝑣))
8 elequ1 2105 . . . . 5 (𝑦 = 𝑤 → (𝑦𝑣𝑤𝑣))
97, 8bi2anan9 636 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝑣𝑦𝑣) ↔ (𝑧𝑣𝑤𝑣)))
109rexbidv 3172 . . 3 ((𝑥 = 𝑧𝑦 = 𝑤) → (∃𝑣𝐴 (𝑥𝑣𝑦𝑣) ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣)))
116, 10bitrid 283 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → (∃𝑢𝐴 (𝑥𝑢𝑦𝑢) ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣)))
12 prtlem13.1 . 2 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
131, 2, 11, 12braba 5530 1 (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wrex 3064   class class class wbr 5141  {copab 5203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204
This theorem is referenced by:  prtlem16  38252  prtlem18  38260  prter1  38262  prter3  38265
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