Theorem vopelopabsb 5513

Description: The law of concretion in terms of substitutions. Version of opelopabsb 5514 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

Step	Hyp	Ref	Expression
1		eqcom 2768	. . . . 5 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑤⟩)
2		vex 3457	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3457	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	opth 5458	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑤⟩ ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))
5	1, 4	bitri 278	. . . 4 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))
6	5	anbi1i 635	. . 3 ⊢ ((⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑))
7	6	2exbii 1877	. 2 ⊢ (∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑))
8		elopab 5511	. 2 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
9		2sb5 2311	. 2 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑))
10	7, 8, 9	3bitr4i 306	1 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1568  ∃wex 1807  [wsb 2094   ∈ wcel 2141  ⟨cop 4594  {copab 5172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-opab 5173

This theorem is referenced by:  opelopabsb  5514  inopab  5816  difopab  5817  isarep1  6624  brabsb2  39582