Theorem elrabsf 1456

Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 1421 has implicit substitution). The hypothesis specifies that x must not be a free variable in B.

Hypothesis

Ref	Expression
elrabsf.1	⊢ (y ∈ B → ∀x y ∈ B)

Assertion

Ref	Expression
elrabsf	⊢ (A ∈ {x ∈ B∣φ} ↔ (A ∈ B ∧ [A / x]φ))

Distinct variable group(s):   y,B   x,y

Proof of Theorem elrabsf

Step	Hyp	Ref	Expression
1		elrabsf.1	. . . 4 ⊢ (y ∈ B → ∀x y ∈ B)
2		ax-17 925	. . . 4 ⊢ (y ∈ B → ∀z y ∈ B)
3		ax-17 925	. . . 4 ⊢ (φ → ∀zφ)
4		hbs1 986	. . . 4 ⊢ ([z / x]φ → ∀x[z / x]φ)
5		sbequ12 865	. . . 4 ⊢ (x = z → (φ ↔ [z / x]φ))
6	1, 2, 3, 4, 5	cbvrab 1425	. . 3 ⊢ {x ∈ B∣φ} = {z ∈ B∣[z / x]φ}
7	6	eleq2i 1153	. 2 ⊢ (A ∈ {x ∈ B∣φ} ↔ A ∈ {z ∈ B∣[z / x]φ})
8		ax-17 925	. . . 4 ⊢ (w ∈ A → ∀z w ∈ A)
9		ax-17 925	. . . 4 ⊢ (w ∈ B → ∀z w ∈ B)
10	8	hbsbc 1446	. . . 4 ⊢ ((A ∈ V → [A / z][z / x]φ) → ∀z(A ∈ V → [A / z][z / x]φ))
11		sbceq1 1443	. . . . 5 ⊢ (z = A → ([z / x]φ ↔ [A / z][z / x]φ))
12		19.8a 712	. . . . . . 7 ⊢ (z = A → ∃z z = A)
13		isset 1351	. . . . . . 7 ⊢ (A ∈ V ↔ ∃z z = A)
14	12, 13	sylibr 175	. . . . . 6 ⊢ (z = A → A ∈ V)
15		biimt 549	. . . . . 6 ⊢ (A ∈ V → ([A / z][z / x]φ ↔ (A ∈ V → [A / z][z / x]φ)))
16	14, 15	syl 12	. . . . 5 ⊢ (z = A → ([A / z][z / x]φ ↔ (A ∈ V → [A / z][z / x]φ)))
17	11, 16	bitrd 406	. . . 4 ⊢ (z = A → ([z / x]φ ↔ (A ∈ V → [A / z][z / x]φ)))
18	8, 9, 10, 17	elrabf 1421	. . 3 ⊢ (A ∈ {z ∈ B∣[z / x]φ} ↔ (A ∈ B ∧ (A ∈ V → [A / z][z / x]φ)))
19		elisset 1354	. . . . 5 ⊢ (A ∈ B → A ∈ V)
20	19, 15	syl 12	. . . 4 ⊢ (A ∈ B → ([A / z][z / x]φ ↔ (A ∈ V → [A / z][z / x]φ)))
21	20	pm5.32i 489	. . 3 ⊢ ((A ∈ B ∧ [A / z][z / x]φ) ↔ (A ∈ B ∧ (A ∈ V → [A / z][z / x]φ)))
22	18, 21	bitr4 154	. 2 ⊢ (A ∈ {z ∈ B∣[z / x]φ} ↔ (A ∈ B ∧ [A / z][z / x]φ))
23		sbcco 1448	. . 3 ⊢ (A ∈ B → ([A / z][z / x]φ ↔ [A / x]φ))
24	23	pm5.32i 489	. 2 ⊢ ((A ∈ B ∧ [A / z][z / x]φ) ↔ (A ∈ B ∧ [A / x]φ))
25	7, 22, 24	3bitr 155	1 ⊢ (A ∈ {x ∈ B∣φ} ↔ (A ∈ B ∧ [A / x]φ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678  [wsb 852   = wceq 1091   ∈ wcel 1092  {crab 1204  Vcvv 1348  [wsbc 1440

This theorem is referenced by:  elabs2 1457  iunrab 2022  tfis 2245  onminesb 2265

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-16 922  ax-17 925  ax-ext 1074

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-rab 1208  df-v 1349  df-sbc 1441

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