Theorem opelopabsb 5410

Description: The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.)

Assertion

Ref	Expression
opelopabsb	⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)

Distinct variable groups:   𝑥,𝑦   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑦)

Proof of Theorem opelopabsb

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3498	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
2		vex 3498	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
3	1, 2	opnzi 5359	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ≠ ∅
4		simpl 485	. . . . . . . . . . 11 ⊢ ((∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∅ = ⟨𝑥, 𝑦⟩)
5	4	eqcomd 2827	. . . . . . . . . 10 ⊢ ((∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ⟨𝑥, 𝑦⟩ = ∅)
6	5	necon3ai 3041	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ≠ ∅ → ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
7	3, 6	ax-mp 5	. . . . . . . 8 ⊢ ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
8	7	nex 1797	. . . . . . 7 ⊢ ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
9	8	nex 1797	. . . . . 6 ⊢ ¬ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
10		elopab 5407	. . . . . 6 ⊢ (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
11	9, 10	mtbir 325	. . . . 5 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
12		eleq1 2900	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ = ∅ → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
13	11, 12	mtbiri 329	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ = ∅ → ¬ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
14	13	necon2ai 3045	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨𝐴, 𝐵⟩ ≠ ∅)
15		opnz 5358	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
16	14, 15	sylib 220	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝐴 ∈ V ∧ 𝐵 ∈ V))
17		sbcex 3782	. . 3 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → 𝐴 ∈ V)
18		spesbc 3865	. . . 4 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → ∃𝑥[𝐵 / 𝑦]𝜑)
19		sbcex 3782	. . . . 5 ⊢ ([𝐵 / 𝑦]𝜑 → 𝐵 ∈ V)
20	19	exlimiv 1927	. . . 4 ⊢ (∃𝑥[𝐵 / 𝑦]𝜑 → 𝐵 ∈ V)
21	18, 20	syl 17	. . 3 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → 𝐵 ∈ V)
22	17, 21	jca 514	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
23		opeq1 4797	. . . . 5 ⊢ (𝑧 = 𝐴 → ⟨𝑧, 𝑤⟩ = ⟨𝐴, 𝑤⟩)
24	23	eleq1d 2897	. . . 4 ⊢ (𝑧 = 𝐴 → (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝐴, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
25		dfsbcq2 3775	. . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝑤 / 𝑦]𝜑))
26	24, 25	bibi12d 348	. . 3 ⊢ (𝑧 = 𝐴 → ((⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ (⟨𝐴, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝑤 / 𝑦]𝜑)))
27		opeq2 4798	. . . . 5 ⊢ (𝑤 = 𝐵 → ⟨𝐴, 𝑤⟩ = ⟨𝐴, 𝐵⟩)
28	27	eleq1d 2897	. . . 4 ⊢ (𝑤 = 𝐵 → (⟨𝐴, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
29		dfsbcq2 3775	. . . . 5 ⊢ (𝑤 = 𝐵 → ([𝑤 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜑))
30	29	sbcbidv 3827	. . . 4 ⊢ (𝑤 = 𝐵 → ([𝐴 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑))
31	28, 30	bibi12d 348	. . 3 ⊢ (𝑤 = 𝐵 → ((⟨𝐴, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝑤 / 𝑦]𝜑) ↔ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)))
32		nfopab1 5128	. . . . . 6 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
33	32	nfel2 2996	. . . . 5 ⊢ Ⅎ𝑥⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
34		nfs1v 2269	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥][𝑤 / 𝑦]𝜑
35	33, 34	nfbi 1900	. . . 4 ⊢ Ⅎ𝑥(⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
36		opeq1 4797	. . . . . 6 ⊢ (𝑥 = 𝑧 → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩)
37	36	eleq1d 2897	. . . . 5 ⊢ (𝑥 = 𝑧 → (⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
38		sbequ12 2248	. . . . 5 ⊢ (𝑥 = 𝑧 → ([𝑤 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
39	37, 38	bibi12d 348	. . . 4 ⊢ (𝑥 = 𝑧 → ((⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦]𝜑) ↔ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)))
40		nfopab2 5129	. . . . . . 7 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
41	40	nfel2 2996	. . . . . 6 ⊢ Ⅎ𝑦⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
42		nfs1v 2269	. . . . . 6 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜑
43	41, 42	nfbi 1900	. . . . 5 ⊢ Ⅎ𝑦(⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦]𝜑)
44		opeq2 4798	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑤⟩)
45	44	eleq1d 2897	. . . . . 6 ⊢ (𝑦 = 𝑤 → (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
46		sbequ12 2248	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑦]𝜑))
47	45, 46	bibi12d 348	. . . . 5 ⊢ (𝑦 = 𝑤 → ((⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑) ↔ (⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦]𝜑)))
48		opabidw 5405	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
49	43, 47, 48	chvarfv 2237	. . . 4 ⊢ (⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦]𝜑)
50	35, 39, 49	chvarfv 2237	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
51	26, 31, 50	vtocl2g 3572	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑))
52	16, 22, 51	pm5.21nii 382	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776  [wsb 2065   ∈ wcel 2110   ≠ wne 3016  Vcvv 3495  [wsbc 3772  ∅c0 4291  ⟨cop 4567  {copab 5121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5122

This theorem is referenced by:  brabsb  5411  opelopabgf  5420  opelopabaf  5424  opelopabf  5425  difopab  5697  isarep1  6437  fmptsnd  6926