Theorem opelopabsb 4182

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 4180	. . . 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 2145	. . . . . . 7 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩ ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ⟨𝑢, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
4		vex 2689	. . . . . . . 8 ⊢ 𝑢 ∈ V
5		vex 2689	. . . . . . . 8 ⊢ 𝑣 ∈ V
6	4, 5	opth 4159	. . . . . . 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 1580	. . . . 5 ⊢ (∃𝑢∃𝑣(⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩ ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ∃𝑢∃𝑣(𝑢 = 𝐴 ∧ 𝑣 = 𝐵))
9		eeanv 1904	. . . . . 6 ⊢ (∃𝑢∃𝑣(𝑢 = 𝐴 ∧ 𝑣 = 𝐵) ↔ (∃𝑢 𝑢 = 𝐴 ∧ ∃𝑣 𝑣 = 𝐵))
10		isset 2692	. . . . . . 7 ⊢ (𝐴 ∈ V ↔ ∃𝑢 𝑢 = 𝐴)
11		isset 2692	. . . . . . 7 ⊢ (𝐵 ∈ V ↔ ∃𝑣 𝑣 = 𝐵)
12	10, 11	anbi12i 455	. . . . . 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 1508	. . . 4 ⊢ Ⅎ𝑢𝜑
17		nfv 1508	. . . 4 ⊢ Ⅎ𝑣𝜑
18		nfs1v 1912	. . . 4 ⊢ Ⅎ𝑥[𝑢 / 𝑥][𝑣 / 𝑦]𝜑
19		nfs1v 1912	. . . . 5 ⊢ Ⅎ𝑦[𝑣 / 𝑦]𝜑
20	19	nfsbxy 1915	. . . 4 ⊢ Ⅎ𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑
21		sbequ12 1744	. . . . 5 ⊢ (𝑦 = 𝑣 → (𝜑 ↔ [𝑣 / 𝑦]𝜑))
22		sbequ12 1744	. . . . 5 ⊢ (𝑥 = 𝑢 → ([𝑣 / 𝑦]𝜑 ↔ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
23	21, 22	sylan9bbr 458	. . . 4 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝜑 ↔ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
24	16, 17, 18, 20, 23	cbvopab 3999	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑢, 𝑣⟩ ∣ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑}
25	15, 24	eleq2s 2234	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝐴 ∈ V ∧ 𝐵 ∈ V))
26		sbcex 2917	. . 3 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → 𝐴 ∈ V)
27		spesbc 2994	. . . 4 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → ∃𝑥[𝐵 / 𝑦]𝜑)
28		sbcex 2917	. . . . 5 ⊢ ([𝐵 / 𝑦]𝜑 → 𝐵 ∈ V)
29	28	exlimiv 1577	. . . 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 3705	. . . . 5 ⊢ (𝑧 = 𝐴 → ⟨𝑧, 𝑤⟩ = ⟨𝐴, 𝑤⟩)
33	32	eleq1d 2208	. . . 4 ⊢ (𝑧 = 𝐴 → (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝐴, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
34		dfsbcq2 2912	. . . 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 3706	. . . . 5 ⊢ (𝑤 = 𝐵 → ⟨𝐴, 𝑤⟩ = ⟨𝐴, 𝐵⟩)
37	36	eleq1d 2208	. . . 4 ⊢ (𝑤 = 𝐵 → (⟨𝐴, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
38		dfsbcq2 2912	. . . . 5 ⊢ (𝑤 = 𝐵 → ([𝑤 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜑))
39	38	sbcbidv 2967	. . . 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 3997	. . . . . 6 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
42	41	nfel2 2294	. . . . 5 ⊢ Ⅎ𝑥⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
43		nfs1v 1912	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥][𝑤 / 𝑦]𝜑
44	42, 43	nfbi 1568	. . . 4 ⊢ Ⅎ𝑥(⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
45		opeq1 3705	. . . . . 6 ⊢ (𝑥 = 𝑧 → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩)
46	45	eleq1d 2208	. . . . 5 ⊢ (𝑥 = 𝑧 → (⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
47		sbequ12 1744	. . . . 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 3998	. . . . . . 7 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
50	49	nfel2 2294	. . . . . 6 ⊢ Ⅎ𝑦⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
51		nfs1v 1912	. . . . . 6 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜑
52	50, 51	nfbi 1568	. . . . 5 ⊢ Ⅎ𝑦(⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦]𝜑)
53		opeq2 3706	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑤⟩)
54	53	eleq1d 2208	. . . . . 6 ⊢ (𝑦 = 𝑤 → (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
55		sbequ12 1744	. . . . . 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 4179	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
58	52, 56, 57	chvar 1730	. . . 4 ⊢ (⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦]𝜑)
59	44, 48, 58	chvar 1730	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
60	35, 40, 59	vtocl2g 2750	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑))
61	25, 31, 60	pm5.21nii 693	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480  [wsb 1735  Vcvv 2686  [wsbc 2909  ⟨cop 3530  {copab 3988

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990

This theorem is referenced by:  brabsb  4183  opelopabaf  4195  opelopabf  4196  difopab  4672  isarep1  5209