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Theorem vopelopabsb 5442
Description: The law of concretion in terms of substitutions. Version of opelopabsb 5443 with set variables. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Remove unnecessary commutation. (Revised by SN, 1-Sep-2024.)
Assertion
Ref Expression
vopelopabsb (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
Distinct variable groups:   𝑥,𝑦,𝑧   𝑥,𝑤,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem vopelopabsb
StepHypRef Expression
1 eqcom 2745 . . . . 5 (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑤⟩)
2 vex 3436 . . . . . 6 𝑥 ∈ V
3 vex 3436 . . . . . 6 𝑦 ∈ V
42, 3opth 5391 . . . . 5 (⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑤⟩ ↔ (𝑥 = 𝑧𝑦 = 𝑤))
51, 4bitri 274 . . . 4 (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝑧𝑦 = 𝑤))
65anbi1i 624 . . 3 ((⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ((𝑥 = 𝑧𝑦 = 𝑤) ∧ 𝜑))
762exbii 1851 . 2 (∃𝑥𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ 𝜑))
8 elopab 5440 . 2 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
9 2sb5 2272 . 2 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ 𝜑))
107, 8, 93bitr4i 303 1 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1539  wex 1782  [wsb 2067  wcel 2106  cop 4567  {copab 5136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5137
This theorem is referenced by:  opelopabsb  5443  inopab  5739  cnvopab  6042  brabsb2  36876
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