Theorem opabid 2099

Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61.

Assertion

Ref	Expression
opabid	⊢ (⟨x, y⟩ ∈ {⟨x, y⟩∣φ} ↔ φ)

Distinct variable group(s):   x,y

Proof of Theorem opabid

Step	Hyp	Ref	Expression
1		clelab 1187	. 2 ⊢ (⟨x, y⟩ ∈ {z∣∃x∃y(z = ⟨x, y⟩ ∧ φ)} ↔ ∃z(z = ⟨x, y⟩ ∧ ∃x∃y(z = ⟨x, y⟩ ∧ φ)))
2		df-opab 2098	. . 3 ⊢ {⟨x, y⟩∣φ} = {z∣∃x∃y(z = ⟨x, y⟩ ∧ φ)}
3	2	eleq2i 1153	. 2 ⊢ (⟨x, y⟩ ∈ {⟨x, y⟩∣φ} ↔ ⟨x, y⟩ ∈ {z∣∃x∃y(z = ⟨x, y⟩ ∧ φ)})
4		19.41v 963	. . . . . 6 ⊢ (∃z((z = ⟨x, y⟩ ∧ z = ⟨w, v⟩) ∧ [v / y][w / x]φ) ↔ (∃z(z = ⟨x, y⟩ ∧ z = ⟨w, v⟩) ∧ [v / y][w / x]φ))
5		anass 336	. . . . . . 7 ⊢ (((z = ⟨x, y⟩ ∧ z = ⟨w, v⟩) ∧ [v / y][w / x]φ) ↔ (z = ⟨x, y⟩ ∧ (z = ⟨w, v⟩ ∧ [v / y][w / x]φ)))
6	5	biex 733	. . . . . 6 ⊢ (∃z((z = ⟨x, y⟩ ∧ z = ⟨w, v⟩) ∧ [v / y][w / x]φ) ↔ ∃z(z = ⟨x, y⟩ ∧ (z = ⟨w, v⟩ ∧ [v / y][w / x]φ)))
7		cleqcom 1103	. . . . . . . 8 ⊢ (⟨x, y⟩ = ⟨w, v⟩ ↔ ⟨w, v⟩ = ⟨x, y⟩)
8		opex 1893	. . . . . . . . 9 ⊢ ⟨x, y⟩ ∈ V
9	8	eqvinc 1407	. . . . . . . 8 ⊢ (⟨x, y⟩ = ⟨w, v⟩ ↔ ∃z(z = ⟨x, y⟩ ∧ z = ⟨w, v⟩))
10		visset 1350	. . . . . . . . 9 ⊢ w ∈ V
11		visset 1350	. . . . . . . . 9 ⊢ v ∈ V
12		visset 1350	. . . . . . . . 9 ⊢ y ∈ V
13	10, 11, 12	opth 1898	. . . . . . . 8 ⊢ (⟨w, v⟩ = ⟨x, y⟩ ↔ (w = x ∧ v = y))
14	7, 9, 13	3bitr3 156	. . . . . . 7 ⊢ (∃z(z = ⟨x, y⟩ ∧ z = ⟨w, v⟩) ↔ (w = x ∧ v = y))
15	14	anbi1i 368	. . . . . 6 ⊢ ((∃z(z = ⟨x, y⟩ ∧ z = ⟨w, v⟩) ∧ [v / y][w / x]φ) ↔ ((w = x ∧ v = y) ∧ [v / y][w / x]φ))
16	4, 6, 15	3bitr3r 157	. . . . 5 ⊢ (((w = x ∧ v = y) ∧ [v / y][w / x]φ) ↔ ∃z(z = ⟨x, y⟩ ∧ (z = ⟨w, v⟩ ∧ [v / y][w / x]φ)))
17	16	bi2ex 734	. . . 4 ⊢ (∃w∃v((w = x ∧ v = y) ∧ [v / y][w / x]φ) ↔ ∃w∃v∃z(z = ⟨x, y⟩ ∧ (z = ⟨w, v⟩ ∧ [v / y][w / x]φ)))
18		sbel2x 995	. . . 4 ⊢ (φ ↔ ∃w∃v((w = x ∧ v = y) ∧ [v / y][w / x]φ))
19		excom 728	. . . . 5 ⊢ (∃z∃w∃v(z = ⟨x, y⟩ ∧ (z = ⟨w, v⟩ ∧ [v / y][w / x]φ)) ↔ ∃w∃z∃v(z = ⟨x, y⟩ ∧ (z = ⟨w, v⟩ ∧ [v / y][w / x]φ)))
20		exdistr2 969	. . . . 5 ⊢ (∃z∃w∃v(z = ⟨x, y⟩ ∧ (z = ⟨w, v⟩ ∧ [v / y][w / x]φ)) ↔ ∃z(z = ⟨x, y⟩ ∧ ∃w∃v(z = ⟨w, v⟩ ∧ [v / y][w / x]φ)))
21		excom 728	. . . . . 6 ⊢ (∃z∃v(z = ⟨x, y⟩ ∧ (z = ⟨w, v⟩ ∧ [v / y][w / x]φ)) ↔ ∃v∃z(z = ⟨x, y⟩ ∧ (z = ⟨w, v⟩ ∧ [v / y][w / x]φ)))
22	21	biex 733	. . . . 5 ⊢ (∃w∃z∃v(z = ⟨x, y⟩ ∧ (z = ⟨w, v⟩ ∧ [v / y][w / x]φ)) ↔ ∃w∃v∃z(z = ⟨x, y⟩ ∧ (z = ⟨w, v⟩ ∧ [v / y][w / x]φ)))
23	19, 20, 22	3bitr3 156	. . . 4 ⊢ (∃z(z = ⟨x, y⟩ ∧ ∃w∃v(z = ⟨w, v⟩ ∧ [v / y][w / x]φ)) ↔ ∃w∃v∃z(z = ⟨x, y⟩ ∧ (z = ⟨w, v⟩ ∧ [v / y][w / x]φ)))
24	17, 18, 23	3bitr4 158	. . 3 ⊢ (φ ↔ ∃z(z = ⟨x, y⟩ ∧ ∃w∃v(z = ⟨w, v⟩ ∧ [v / y][w / x]φ)))
25		ax-17 925	. . . . . . 7 ⊢ (∃y(z = ⟨x, y⟩ ∧ φ) → ∀w∃y(z = ⟨x, y⟩ ∧ φ))
26		ax-17 925	. . . . . . . . 9 ⊢ (z = ⟨w, y⟩ → ∀x z = ⟨w, y⟩)
27		hbs1 986	. . . . . . . . 9 ⊢ ([w / x]φ → ∀x[w / x]φ)
28	26, 27	hban 704	. . . . . . . 8 ⊢ ((z = ⟨w, y⟩ ∧ [w / x]φ) → ∀x(z = ⟨w, y⟩ ∧ [w / x]φ))
29	28	hbex 701	. . . . . . 7 ⊢ (∃y(z = ⟨w, y⟩ ∧ [w / x]φ) → ∀x∃y(z = ⟨w, y⟩ ∧ [w / x]φ))
30		opeq1 1876	. . . . . . . . . 10 ⊢ (x = w → ⟨x, y⟩ = ⟨w, y⟩)
31	30	cleq2d 1112	. . . . . . . . 9 ⊢ (x = w → (z = ⟨x, y⟩ ↔ z = ⟨w, y⟩))
32		sbequ12 865	. . . . . . . . 9 ⊢ (x = w → (φ ↔ [w / x]φ))
33	31, 32	anbi12d 476	. . . . . . . 8 ⊢ (x = w → ((z = ⟨x, y⟩ ∧ φ) ↔ (z = ⟨w, y⟩ ∧ [w / x]φ)))
34	33	biexdv 936	. . . . . . 7 ⊢ (x = w → (∃y(z = ⟨x, y⟩ ∧ φ) ↔ ∃y(z = ⟨w, y⟩ ∧ [w / x]φ)))
35	25, 29, 34	cbvex 849	. . . . . 6 ⊢ (∃x∃y(z = ⟨x, y⟩ ∧ φ) ↔ ∃w∃y(z = ⟨w, y⟩ ∧ [w / x]φ))
36		ax-17 925	. . . . . . . 8 ⊢ ((z = ⟨w, y⟩ ∧ [w / x]φ) → ∀v(z = ⟨w, y⟩ ∧ [w / x]φ))
37		ax-17 925	. . . . . . . . 9 ⊢ (z = ⟨w, v⟩ → ∀y z = ⟨w, v⟩)
38		hbs1 986	. . . . . . . . 9 ⊢ ([v / y][w / x]φ → ∀y[v / y][w / x]φ)
39	37, 38	hban 704	. . . . . . . 8 ⊢ ((z = ⟨w, v⟩ ∧ [v / y][w / x]φ) → ∀y(z = ⟨w, v⟩ ∧ [v / y][w / x]φ))
40		opeq2 1877	. . . . . . . . . 10 ⊢ (y = v → ⟨w, y⟩ = ⟨w, v⟩)
41	40	cleq2d 1112	. . . . . . . . 9 ⊢ (y = v → (z = ⟨w, y⟩ ↔ z = ⟨w, v⟩))
42		sbequ12 865	. . . . . . . . 9 ⊢ (y = v → ([w / x]φ ↔ [v / y][w / x]φ))
43	41, 42	anbi12d 476	. . . . . . . 8 ⊢ (y = v → ((z = ⟨w, y⟩ ∧ [w / x]φ) ↔ (z = ⟨w, v⟩ ∧ [v / y][w / x]φ)))
44	36, 39, 43	cbvex 849	. . . . . . 7 ⊢ (∃y(z = ⟨w, y⟩ ∧ [w / x]φ) ↔ ∃v(z = ⟨w, v⟩ ∧ [v / y][w / x]φ))
45	44	biex 733	. . . . . 6 ⊢ (∃w∃y(z = ⟨w, y⟩ ∧ [w / x]φ) ↔ ∃w∃v(z = ⟨w, v⟩ ∧ [v / y][w / x]φ))
46	35, 45	bitr 151	. . . . 5 ⊢ (∃x∃y(z = ⟨x, y⟩ ∧ φ) ↔ ∃w∃v(z = ⟨w, v⟩ ∧ [v / y][w / x]φ))
47	46	anbi2i 367	. . . 4 ⊢ ((z = ⟨x, y⟩ ∧ ∃x∃y(z = ⟨x, y⟩ ∧ φ)) ↔ (z = ⟨x, y⟩ ∧ ∃w∃v(z = ⟨w, v⟩ ∧ [v / y][w / x]φ)))
48	47	biex 733	. . 3 ⊢ (∃z(z = ⟨x, y⟩ ∧ ∃x∃y(z = ⟨x, y⟩ ∧ φ)) ↔ ∃z(z = ⟨x, y⟩ ∧ ∃w∃v(z = ⟨w, v⟩ ∧ [v / y][w / x]φ)))
49	24, 48	bitr4 154	. 2 ⊢ (φ ↔ ∃z(z = ⟨x, y⟩ ∧ ∃x∃y(z = ⟨x, y⟩ ∧ φ)))
50	1, 3, 49	3bitr4 158	1 ⊢ (⟨x, y⟩ ∈ {⟨x, y⟩∣φ} ↔ φ)

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   = weq 797  [wsb 852  {cab 1090   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  {copab 2055

This theorem is referenced by:  opabsb 2114  opelopabg 2115  dmopab 2539  rnopab 2566  cotr 2625  cnvsym 2626  cnvopab 2632  funopab 2694  zfrep6 2744  fnopabg 2745  fvopab2 2878  enssdom 3287

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098

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