Theorem elrabsf 1966

Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 1907 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 groups:   y,B   x,y

Proof of Theorem elrabsf

Step	Hyp	Ref	Expression
1		elrabsf.1	. . . 4 ⊢ (y ∈ B → ∀x y ∈ B)
2		ax-17 973	. . . 4 ⊢ (y ∈ B → ∀z y ∈ B)
3		ax-17 973	. . . 4 ⊢ (φ → ∀zφ)
4		hbs1 1334	. . . 4 ⊢ ([z / x]φ → ∀x[z / x]φ)
5		sbequ12 1183	. . . 4 ⊢ (x = z → (φ ↔ [z / x]φ))
6	1, 2, 3, 4, 5	cbvrab 1913	. . 3 ⊢ {x ∈ B∣φ} = {z ∈ B∣[z / x]φ}
7	6	eleq2i 1541	. 2 ⊢ (A ∈ {x ∈ B∣φ} ↔ A ∈ {z ∈ B∣[z / x]φ})
8		ax-17 973	. . . 4 ⊢ (w ∈ A → ∀z w ∈ A)
9		ax-17 973	. . . 4 ⊢ (w ∈ B → ∀z w ∈ B)
10	8	hbsbc1 1952	. . . 4 ⊢ ((A ∈ V → [A / z][z / x]φ) → ∀z(A ∈ V → [A / z][z / x]φ))
11		sbceq1a 1947	. . . . 5 ⊢ (z = A → ([z / x]φ ↔ [A / z][z / x]φ))
12		19.8a 1031	. . . . . . 7 ⊢ (z = A → ∃z z = A)
13		isset 1817	. . . . . . 7 ⊢ (A ∈ V ↔ ∃z z = A)
14	12, 13	sylibr 200	. . . . . 6 ⊢ (z = A → A ∈ V)
15		biimt 733	. . . . . 6 ⊢ (A ∈ V → ([A / z][z / x]φ ↔ (A ∈ V → [A / z][z / x]φ)))
16	14, 15	syl 10	. . . . 5 ⊢ (z = A → ([A / z][z / x]φ ↔ (A ∈ V → [A / z][z / x]φ)))
17	11, 16	bitrd 530	. . . 4 ⊢ (z = A → ([z / x]φ ↔ (A ∈ V → [A / z][z / x]φ)))
18	8, 9, 10, 17	elrabf 1907	. . 3 ⊢ (A ∈ {z ∈ B∣[z / x]φ} ↔ (A ∈ B ⋀ (A ∈ V → [A / z][z / x]φ)))
19		elisset 1820	. . . . 5 ⊢ (A ∈ B → A ∈ V)
20	19, 15	syl 10	. . . 4 ⊢ (A ∈ B → ([A / z][z / x]φ ↔ (A ∈ V → [A / z][z / x]φ)))
21	20	pm5.32i 647	. . 3 ⊢ ((A ∈ B ⋀ [A / z][z / x]φ) ↔ (A ∈ B ⋀ (A ∈ V → [A / z][z / x]φ)))
22	18, 21	bitr4 176	. 2 ⊢ (A ∈ {z ∈ B∣[z / x]φ} ↔ (A ∈ B ⋀ [A / z][z / x]φ))
23		sbccog 1955	. . 3 ⊢ (A ∈ B → ([A / z][z / x]φ ↔ [A / x]φ))
24	23	pm5.32i 647	. 2 ⊢ ((A ∈ B ⋀ [A / z][z / x]φ) ↔ (A ∈ B ⋀ [A / x]φ))
25	7, 22, 24	3bitr 177	1 ⊢ (A ∈ {x ∈ B∣φ} ↔ (A ∈ B ⋀ [A / x]φ))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 956   = wceq 958   ∈ wcel 960  ∃wex 982  [wsbc 1172  {crab 1651  Vcvv 1814

This theorem is referenced by:  elabs2 1967  iunrab 2600  reucl2 2894  onminesb 3016  tfis 3133

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-rab 1655  df-v 1815  df-sbc 1945

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