Theorem opabsb 2114

Description: The law of concretion in terms of substitutions.

Assertion

Ref	Expression
opabsb	⊢ (⟨z, w⟩ ∈ {⟨x, y⟩∣φ} ↔ [w / y][z / x]φ)

Distinct variable group(s):   x,y,z,w

Proof of Theorem opabsb

Step	Hyp	Ref	Expression
1		a9e 809	. 2 ⊢ ∃y y = w
2		ax-17 925	. . . . 5 ⊢ (v ∈ ⟨z, w⟩ → ∀y v ∈ ⟨z, w⟩)
3		hbopab2 2113	. . . . 5 ⊢ (v ∈ {⟨x, y⟩∣φ} → ∀y v ∈ {⟨x, y⟩∣φ})
4	2, 3	hbel 1172	. . . 4 ⊢ (⟨z, w⟩ ∈ {⟨x, y⟩∣φ} → ∀y⟨z, w⟩ ∈ {⟨x, y⟩∣φ})
5		hbs1 986	. . . 4 ⊢ ([w / y][z / x]φ → ∀y[w / y][z / x]φ)
6	4, 5	hbbi 705	. . 3 ⊢ ((⟨z, w⟩ ∈ {⟨x, y⟩∣φ} ↔ [w / y][z / x]φ) → ∀y(⟨z, w⟩ ∈ {⟨x, y⟩∣φ} ↔ [w / y][z / x]φ))
7		a9e 809	. . . 4 ⊢ ∃x x = z
8		ax-17 925	. . . . . 6 ⊢ (y = w → ∀x y = w)
9		ax-17 925	. . . . . . . 8 ⊢ (v ∈ ⟨z, w⟩ → ∀x v ∈ ⟨z, w⟩)
10		hbopab1 2112	. . . . . . . 8 ⊢ (v ∈ {⟨x, y⟩∣φ} → ∀x v ∈ {⟨x, y⟩∣φ})
11	9, 10	hbel 1172	. . . . . . 7 ⊢ (⟨z, w⟩ ∈ {⟨x, y⟩∣φ} → ∀x⟨z, w⟩ ∈ {⟨x, y⟩∣φ})
12		hbs1 986	. . . . . . . 8 ⊢ ([z / x]φ → ∀x[z / x]φ)
13	12	hbsb 987	. . . . . . 7 ⊢ ([w / y][z / x]φ → ∀x[w / y][z / x]φ)
14	11, 13	hbbi 705	. . . . . 6 ⊢ ((⟨z, w⟩ ∈ {⟨x, y⟩∣φ} ↔ [w / y][z / x]φ) → ∀x(⟨z, w⟩ ∈ {⟨x, y⟩∣φ} ↔ [w / y][z / x]φ))
15	8, 14	hbim 702	. . . . 5 ⊢ ((y = w → (⟨z, w⟩ ∈ {⟨x, y⟩∣φ} ↔ [w / y][z / x]φ)) → ∀x(y = w → (⟨z, w⟩ ∈ {⟨x, y⟩∣φ} ↔ [w / y][z / x]φ)))
16		opeq12 1878	. . . . . . . . 9 ⊢ ((x = z ∧ y = w) → ⟨x, y⟩ = ⟨z, w⟩)
17	16	eleq1d 1155	. . . . . . . 8 ⊢ ((x = z ∧ y = w) → (⟨x, y⟩ ∈ {⟨x, y⟩∣φ} ↔ ⟨z, w⟩ ∈ {⟨x, y⟩∣φ}))
18		opabid 2099	. . . . . . . 8 ⊢ (⟨x, y⟩ ∈ {⟨x, y⟩∣φ} ↔ φ)
19	17, 18	syl5bbr 412	. . . . . . 7 ⊢ ((x = z ∧ y = w) → (φ ↔ ⟨z, w⟩ ∈ {⟨x, y⟩∣φ}))
20		sbequ12 865	. . . . . . . 8 ⊢ (x = z → (φ ↔ [z / x]φ))
21		sbequ12 865	. . . . . . . 8 ⊢ (y = w → ([z / x]φ ↔ [w / y][z / x]φ))
22	20, 21	sylan9bb 418	. . . . . . 7 ⊢ ((x = z ∧ y = w) → (φ ↔ [w / y][z / x]φ))
23	19, 22	bitr3d 408	. . . . . 6 ⊢ ((x = z ∧ y = w) → (⟨z, w⟩ ∈ {⟨x, y⟩∣φ} ↔ [w / y][z / x]φ))
24	23	exp 291	. . . . 5 ⊢ (x = z → (y = w → (⟨z, w⟩ ∈ {⟨x, y⟩∣φ} ↔ [w / y][z / x]φ)))
25	15, 24	19.23ai 746	. . . 4 ⊢ (∃x x = z → (y = w → (⟨z, w⟩ ∈ {⟨x, y⟩∣φ} ↔ [w / y][z / x]φ)))
26	7, 25	ax-mp 6	. . 3 ⊢ (y = w → (⟨z, w⟩ ∈ {⟨x, y⟩∣φ} ↔ [w / y][z / x]φ))
27	6, 26	19.23ai 746	. 2 ⊢ (∃y y = w → (⟨z, w⟩ ∈ {⟨x, y⟩∣φ} ↔ [w / y][z / x]φ))
28	1, 27	ax-mp 6	1 ⊢ (⟨z, w⟩ ∈ {⟨x, y⟩∣φ} ↔ [w / y][z / x]φ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = weq 797  [wsb 852   ∈ wcel 1092  ⟨cop 1810  {copab 2055

This theorem is referenced by:  inopab 2495

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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