Theorem opelopabsb 4245

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 𝑣 𝑢 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elopab 4243	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑} ↔ ∃𝑢∃𝑣(⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩ ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
2		simpl 108	. . . . . . . 8 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩ ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩)
3	2	eqcomd 2176	. . . . . . 7 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩ ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ⟨𝑢, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
4		vex 2733	. . . . . . . 8 ⊢ 𝑢 ∈ V
5		vex 2733	. . . . . . . 8 ⊢ 𝑣 ∈ V
6	4, 5	opth 4222	. . . . . . 7 ⊢ (⟨𝑢, 𝑣⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑢 = 𝐴 ∧ 𝑣 = 𝐵))
7	3, 6	sylib 121	. . . . . 6 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩ ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → (𝑢 = 𝐴 ∧ 𝑣 = 𝐵))
8	7	2eximi 1594	. . . . 5 ⊢ (∃𝑢∃𝑣(⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩ ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ∃𝑢∃𝑣(𝑢 = 𝐴 ∧ 𝑣 = 𝐵))
9		eeanv 1925	. . . . . 6 ⊢ (∃𝑢∃𝑣(𝑢 = 𝐴 ∧ 𝑣 = 𝐵) ↔ (∃𝑢 𝑢 = 𝐴 ∧ ∃𝑣 𝑣 = 𝐵))
10		isset 2736	. . . . . . 7 ⊢ (𝐴 ∈ V ↔ ∃𝑢 𝑢 = 𝐴)
11		isset 2736	. . . . . . 7 ⊢ (𝐵 ∈ V ↔ ∃𝑣 𝑣 = 𝐵)
12	10, 11	anbi12i 457	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (∃𝑢 𝑢 = 𝐴 ∧ ∃𝑣 𝑣 = 𝐵))
13	9, 12	bitr4i 186	. . . . 5 ⊢ (∃𝑢∃𝑣(𝑢 = 𝐴 ∧ 𝑣 = 𝐵) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
14	8, 13	sylib 121	. . . 4 ⊢ (∃𝑢∃𝑣(⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩ ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
15	1, 14	sylbi 120	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑} → (𝐴 ∈ V ∧ 𝐵 ∈ V))
16		nfv 1521	. . . 4 ⊢ Ⅎ𝑢𝜑
17		nfv 1521	. . . 4 ⊢ Ⅎ𝑣𝜑
18		nfs1v 1932	. . . 4 ⊢ Ⅎ𝑥[𝑢 / 𝑥][𝑣 / 𝑦]𝜑
19		nfs1v 1932	. . . . 5 ⊢ Ⅎ𝑦[𝑣 / 𝑦]𝜑
20	19	nfsbxy 1935	. . . 4 ⊢ Ⅎ𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑
21		sbequ12 1764	. . . . 5 ⊢ (𝑦 = 𝑣 → (𝜑 ↔ [𝑣 / 𝑦]𝜑))
22		sbequ12 1764	. . . . 5 ⊢ (𝑥 = 𝑢 → ([𝑣 / 𝑦]𝜑 ↔ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
23	21, 22	sylan9bbr 460	. . . 4 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝜑 ↔ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
24	16, 17, 18, 20, 23	cbvopab 4060	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑢, 𝑣⟩ ∣ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑}
25	15, 24	eleq2s 2265	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝐴 ∈ V ∧ 𝐵 ∈ V))
26		sbcex 2963	. . 3 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → 𝐴 ∈ V)
27		spesbc 3040	. . . 4 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → ∃𝑥[𝐵 / 𝑦]𝜑)
28		sbcex 2963	. . . . 5 ⊢ ([𝐵 / 𝑦]𝜑 → 𝐵 ∈ V)
29	28	exlimiv 1591	. . . 4 ⊢ (∃𝑥[𝐵 / 𝑦]𝜑 → 𝐵 ∈ V)
30	27, 29	syl 14	. . 3 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → 𝐵 ∈ V)
31	26, 30	jca 304	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
32		opeq1 3765	. . . . 5 ⊢ (𝑧 = 𝐴 → ⟨𝑧, 𝑤⟩ = ⟨𝐴, 𝑤⟩)
33	32	eleq1d 2239	. . . 4 ⊢ (𝑧 = 𝐴 → (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝐴, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
34		dfsbcq2 2958	. . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝑤 / 𝑦]𝜑))
35	33, 34	bibi12d 234	. . 3 ⊢ (𝑧 = 𝐴 → ((⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ (⟨𝐴, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝑤 / 𝑦]𝜑)))
36		opeq2 3766	. . . . 5 ⊢ (𝑤 = 𝐵 → ⟨𝐴, 𝑤⟩ = ⟨𝐴, 𝐵⟩)
37	36	eleq1d 2239	. . . 4 ⊢ (𝑤 = 𝐵 → (⟨𝐴, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
38		dfsbcq2 2958	. . . . 5 ⊢ (𝑤 = 𝐵 → ([𝑤 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜑))
39	38	sbcbidv 3013	. . . 4 ⊢ (𝑤 = 𝐵 → ([𝐴 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑))
40	37, 39	bibi12d 234	. . 3 ⊢ (𝑤 = 𝐵 → ((⟨𝐴, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝑤 / 𝑦]𝜑) ↔ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)))
41		nfopab1 4058	. . . . . 6 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
42	41	nfel2 2325	. . . . 5 ⊢ Ⅎ𝑥⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
43		nfs1v 1932	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥][𝑤 / 𝑦]𝜑
44	42, 43	nfbi 1582	. . . 4 ⊢ Ⅎ𝑥(⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
45		opeq1 3765	. . . . . 6 ⊢ (𝑥 = 𝑧 → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩)
46	45	eleq1d 2239	. . . . 5 ⊢ (𝑥 = 𝑧 → (⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
47		sbequ12 1764	. . . . 5 ⊢ (𝑥 = 𝑧 → ([𝑤 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
48	46, 47	bibi12d 234	. . . 4 ⊢ (𝑥 = 𝑧 → ((⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦]𝜑) ↔ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)))
49		nfopab2 4059	. . . . . . 7 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
50	49	nfel2 2325	. . . . . 6 ⊢ Ⅎ𝑦⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
51		nfs1v 1932	. . . . . 6 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜑
52	50, 51	nfbi 1582	. . . . 5 ⊢ Ⅎ𝑦(⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦]𝜑)
53		opeq2 3766	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑤⟩)
54	53	eleq1d 2239	. . . . . 6 ⊢ (𝑦 = 𝑤 → (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
55		sbequ12 1764	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑦]𝜑))
56	54, 55	bibi12d 234	. . . . 5 ⊢ (𝑦 = 𝑤 → ((⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑) ↔ (⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦]𝜑)))
57		opabid 4242	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
58	52, 56, 57	chvar 1750	. . . 4 ⊢ (⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦]𝜑)
59	44, 48, 58	chvar 1750	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
60	35, 40, 59	vtocl2g 2794	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑))
61	25, 31, 60	pm5.21nii 699	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1348  ∃wex 1485  [wsb 1755   ∈ wcel 2141  Vcvv 2730  [wsbc 2955  ⟨cop 3586  {copab 4049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051

This theorem is referenced by:  brabsb  4246  opelopabaf  4258  opelopabf  4259  difopab  4744  isarep1  5284