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Theorem opelopabsbALT 4982
Description: The law of concretion in terms of substitutions. Less general than opelopabsb 4983, but having a much shorter proof. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
opelopabsbALT (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
Distinct variable groups:   𝑥,𝑦,𝑧   𝑥,𝑤,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem opelopabsbALT
StepHypRef Expression
1 excom 2041 . . 3 (∃𝑥𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦𝑥(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2 vex 3201 . . . . . . 7 𝑧 ∈ V
3 vex 3201 . . . . . . 7 𝑤 ∈ V
42, 3opth 4943 . . . . . 6 (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑧 = 𝑥𝑤 = 𝑦))
5 equcom 1944 . . . . . . 7 (𝑧 = 𝑥𝑥 = 𝑧)
6 equcom 1944 . . . . . . 7 (𝑤 = 𝑦𝑦 = 𝑤)
75, 6anbi12ci 734 . . . . . 6 ((𝑧 = 𝑥𝑤 = 𝑦) ↔ (𝑦 = 𝑤𝑥 = 𝑧))
84, 7bitri 264 . . . . 5 (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑦 = 𝑤𝑥 = 𝑧))
98anbi1i 731 . . . 4 ((⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ((𝑦 = 𝑤𝑥 = 𝑧) ∧ 𝜑))
1092exbii 1774 . . 3 (∃𝑦𝑥(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦𝑥((𝑦 = 𝑤𝑥 = 𝑧) ∧ 𝜑))
111, 10bitri 264 . 2 (∃𝑥𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦𝑥((𝑦 = 𝑤𝑥 = 𝑧) ∧ 𝜑))
12 elopab 4981 . 2 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
13 2sb5 2442 . 2 ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ ∃𝑦𝑥((𝑦 = 𝑤𝑥 = 𝑧) ∧ 𝜑))
1411, 12, 133bitr4i 292 1 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1482  wex 1703  [wsb 1879  wcel 1989  cop 4181  {copab 4710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-opab 4711
This theorem is referenced by:  inopab  5250  cnvopab  5531  brabsb2  33973
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