Theorem opelopabsb 4397

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 4395	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑} ↔ ∃𝑢∃𝑣(⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩ ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
2		simpl 109	. . . . . . . 8 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩ ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩)
3	2	eqcomd 2244	. . . . . . 7 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩ ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ⟨𝑢, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
4		vex 2824	. . . . . . . 8 ⊢ 𝑢 ∈ V
5		vex 2824	. . . . . . . 8 ⊢ 𝑣 ∈ V
6	4, 5	opth 4372	. . . . . . 7 ⊢ (⟨𝑢, 𝑣⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑢 = 𝐴 ∧ 𝑣 = 𝐵))
7	3, 6	sylib 122	. . . . . 6 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩ ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → (𝑢 = 𝐴 ∧ 𝑣 = 𝐵))
8	7	2eximi 1654	. . . . 5 ⊢ (∃𝑢∃𝑣(⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩ ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ∃𝑢∃𝑣(𝑢 = 𝐴 ∧ 𝑣 = 𝐵))
9		eeanv 1992	. . . . . 6 ⊢ (∃𝑢∃𝑣(𝑢 = 𝐴 ∧ 𝑣 = 𝐵) ↔ (∃𝑢 𝑢 = 𝐴 ∧ ∃𝑣 𝑣 = 𝐵))
10		isset 2828	. . . . . . 7 ⊢ (𝐴 ∈ V ↔ ∃𝑢 𝑢 = 𝐴)
11		isset 2828	. . . . . . 7 ⊢ (𝐵 ∈ V ↔ ∃𝑣 𝑣 = 𝐵)
12	10, 11	anbi12i 464	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (∃𝑢 𝑢 = 𝐴 ∧ ∃𝑣 𝑣 = 𝐵))
13	9, 12	bitr4i 187	. . . . 5 ⊢ (∃𝑢∃𝑣(𝑢 = 𝐴 ∧ 𝑣 = 𝐵) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
14	8, 13	sylib 122	. . . 4 ⊢ (∃𝑢∃𝑣(⟨𝐴, 𝐵⟩ = ⟨𝑢, 𝑣⟩ ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
15	1, 14	sylbi 121	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑} → (𝐴 ∈ V ∧ 𝐵 ∈ V))
16		nfv 1581	. . . 4 ⊢ Ⅎ𝑢𝜑
17		nfv 1581	. . . 4 ⊢ Ⅎ𝑣𝜑
18		nfs1v 1999	. . . 4 ⊢ Ⅎ𝑥[𝑢 / 𝑥][𝑣 / 𝑦]𝜑
19		nfs1v 1999	. . . . 5 ⊢ Ⅎ𝑦[𝑣 / 𝑦]𝜑
20	19	nfsbxy 2002	. . . 4 ⊢ Ⅎ𝑦[𝑢 / 𝑥][𝑣 / 𝑦]𝜑
21		sbequ12 1824	. . . . 5 ⊢ (𝑦 = 𝑣 → (𝜑 ↔ [𝑣 / 𝑦]𝜑))
22		sbequ12 1824	. . . . 5 ⊢ (𝑥 = 𝑢 → ([𝑣 / 𝑦]𝜑 ↔ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
23	21, 22	sylan9bbr 467	. . . 4 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝜑 ↔ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
24	16, 17, 18, 20, 23	cbvopab 4197	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑢, 𝑣⟩ ∣ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑}
25	15, 24	eleq2s 2333	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝐴 ∈ V ∧ 𝐵 ∈ V))
26		sbcex 3060	. . 3 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → 𝐴 ∈ V)
27		spesbc 3138	. . . 4 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → ∃𝑥[𝐵 / 𝑦]𝜑)
28		sbcex 3060	. . . . 5 ⊢ ([𝐵 / 𝑦]𝜑 → 𝐵 ∈ V)
29	28	exlimiv 1651	. . . 4 ⊢ (∃𝑥[𝐵 / 𝑦]𝜑 → 𝐵 ∈ V)
30	27, 29	syl 14	. . 3 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → 𝐵 ∈ V)
31	26, 30	jca 306	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
32		opeq1 3899	. . . . 5 ⊢ (𝑧 = 𝐴 → ⟨𝑧, 𝑤⟩ = ⟨𝐴, 𝑤⟩)
33	32	eleq1d 2307	. . . 4 ⊢ (𝑧 = 𝐴 → (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝐴, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
34		dfsbcq2 3054	. . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝑤 / 𝑦]𝜑))
35	33, 34	bibi12d 235	. . 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 3900	. . . . 5 ⊢ (𝑤 = 𝐵 → ⟨𝐴, 𝑤⟩ = ⟨𝐴, 𝐵⟩)
37	36	eleq1d 2307	. . . 4 ⊢ (𝑤 = 𝐵 → (⟨𝐴, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
38		dfsbcq2 3054	. . . . 5 ⊢ (𝑤 = 𝐵 → ([𝑤 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜑))
39	38	sbcbidv 3110	. . . 4 ⊢ (𝑤 = 𝐵 → ([𝐴 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑))
40	37, 39	bibi12d 235	. . 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 4195	. . . . . 6 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
42	41	nfel2 2405	. . . . 5 ⊢ Ⅎ𝑥⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
43		nfs1v 1999	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥][𝑤 / 𝑦]𝜑
44	42, 43	nfbi 1642	. . . 4 ⊢ Ⅎ𝑥(⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
45		opeq1 3899	. . . . . 6 ⊢ (𝑥 = 𝑧 → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩)
46	45	eleq1d 2307	. . . . 5 ⊢ (𝑥 = 𝑧 → (⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
47		sbequ12 1824	. . . . 5 ⊢ (𝑥 = 𝑧 → ([𝑤 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
48	46, 47	bibi12d 235	. . . 4 ⊢ (𝑥 = 𝑧 → ((⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦]𝜑) ↔ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)))
49		nfopab2 4196	. . . . . . 7 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
50	49	nfel2 2405	. . . . . 6 ⊢ Ⅎ𝑦⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
51		nfs1v 1999	. . . . . 6 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜑
52	50, 51	nfbi 1642	. . . . 5 ⊢ Ⅎ𝑦(⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦]𝜑)
53		opeq2 3900	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑤⟩)
54	53	eleq1d 2307	. . . . . 6 ⊢ (𝑦 = 𝑤 → (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
55		sbequ12 1824	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑦]𝜑))
56	54, 55	bibi12d 235	. . . . 5 ⊢ (𝑦 = 𝑤 → ((⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑) ↔ (⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦]𝜑)))
57		opabid 4393	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
58	52, 56, 57	chvar 1810	. . . 4 ⊢ (⟨𝑥, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦]𝜑)
59	44, 48, 58	chvar 1810	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
60	35, 40, 59	vtocl2g 2887	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑))
61	25, 31, 60	pm5.21nii 716	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1402  ∃wex 1545  [wsb 1815   ∈ wcel 2209  Vcvv 2821  [wsbc 3051  ⟨cop 3708  {copab 4186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-opab 4188

This theorem is referenced by:  brabsb  4398  opelopabgf  4407  opelopabaf  4411  opelopabf  4412  difopab  4908  isarep1  5462