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Theorem prtlem13 37135
Description: Lemma for prter1 37146, prter2 37148, prter3 37149 and prtex 37147. (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 3445 . 2 𝑧 ∈ V
2 vex 3445 . 2 𝑤 ∈ V
3 elequ2 2120 . . . . 5 (𝑢 = 𝑣 → (𝑥𝑢𝑥𝑣))
4 elequ2 2120 . . . . 5 (𝑢 = 𝑣 → (𝑦𝑢𝑦𝑣))
53, 4anbi12d 631 . . . 4 (𝑢 = 𝑣 → ((𝑥𝑢𝑦𝑢) ↔ (𝑥𝑣𝑦𝑣)))
65cbvrexvw 3222 . . 3 (∃𝑢𝐴 (𝑥𝑢𝑦𝑢) ↔ ∃𝑣𝐴 (𝑥𝑣𝑦𝑣))
7 elequ1 2112 . . . . 5 (𝑥 = 𝑧 → (𝑥𝑣𝑧𝑣))
8 elequ1 2112 . . . . 5 (𝑦 = 𝑤 → (𝑦𝑣𝑤𝑣))
97, 8bi2anan9 636 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝑣𝑦𝑣) ↔ (𝑧𝑣𝑤𝑣)))
109rexbidv 3171 . . 3 ((𝑥 = 𝑧𝑦 = 𝑤) → (∃𝑣𝐴 (𝑥𝑣𝑦𝑣) ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣)))
116, 10bitrid 282 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → (∃𝑢𝐴 (𝑥𝑢𝑦𝑢) ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣)))
12 prtlem13.1 . 2 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
131, 2, 11, 12braba 5481 1 (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1540  wrex 3070   class class class wbr 5092  {copab 5154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-br 5093  df-opab 5155
This theorem is referenced by:  prtlem16  37136  prtlem18  37144  prter1  37146  prter3  37149
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