Theorem vopelopabsb 5533

Description: The law of concretion in terms of substitutions. Version of opelopabsb 5534 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 2743	. . . . 5 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑤⟩)
2		vex 3483	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3483	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	opth 5480	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑤⟩ ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))
5	1, 4	bitri 275	. . . 4 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))
6	5	anbi1i 624	. . 3 ⊢ ((⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑))
7	6	2exbii 1848	. 2 ⊢ (∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑))
8		elopab 5531	. 2 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
9		2sb5 2277	. 2 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑))
10	7, 8, 9	3bitr4i 303	1 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1539  ∃wex 1778  [wsb 2063   ∈ wcel 2107  ⟨cop 4631  {copab 5204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-opab 5205

This theorem is referenced by:  opelopabsb  5534  inopab  5838  difopab  5839  cnvopabOLD  6157  isarep1  6655  brabsb2  38864