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Theorem opelopabsbALT 4909
 Description: The law of concretion in terms of substitutions. Less general than opelopabsb 4910, 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 2029 . . 3 (∃𝑥𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦𝑥(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2 vex 3176 . . . . . . 7 𝑧 ∈ V
3 vex 3176 . . . . . . 7 𝑤 ∈ V
42, 3opth 4871 . . . . . 6 (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑧 = 𝑥𝑤 = 𝑦))
5 equcom 1932 . . . . . . 7 (𝑧 = 𝑥𝑥 = 𝑧)
6 equcom 1932 . . . . . . 7 (𝑤 = 𝑦𝑦 = 𝑤)
75, 6anbi12ci 730 . . . . . 6 ((𝑧 = 𝑥𝑤 = 𝑦) ↔ (𝑦 = 𝑤𝑥 = 𝑧))
84, 7bitri 263 . . . . 5 (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑦 = 𝑤𝑥 = 𝑧))
98anbi1i 727 . . . 4 ((⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ((𝑦 = 𝑤𝑥 = 𝑧) ∧ 𝜑))
1092exbii 1765 . . 3 (∃𝑦𝑥(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦𝑥((𝑦 = 𝑤𝑥 = 𝑧) ∧ 𝜑))
111, 10bitri 263 . 2 (∃𝑥𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦𝑥((𝑦 = 𝑤𝑥 = 𝑧) ∧ 𝜑))
12 elopab 4908 . 2 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
13 2sb5 2431 . 2 ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ ∃𝑦𝑥((𝑦 = 𝑤𝑥 = 𝑧) ∧ 𝜑))
1411, 12, 133bitr4i 291 1 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695  [wsb 1867   ∈ wcel 1977  ⟨cop 4131  {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-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  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-opab 4644 This theorem is referenced by:  inopab  5174  cnvopab  5452  brabsb2  33165
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