Theorem elrabsf 3085

Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2994 has implicit substitution). The hypothesis specifies that x must not be a free variable in B. (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Hypothesis

Ref	Expression
elrabsf.1	⊢ ℲxB

Assertion

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

Proof of Theorem elrabsf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq 3049	. 2 ⊢ (y = A → ([̣y / x]̣φ ↔ [̣A / x]̣φ))
2		elrabsf.1	. . 3 ⊢ ℲxB
3		nfcv 2490	. . 3 ⊢ ℲyB
4		nfv 1619	. . 3 ⊢ Ⅎyφ
5		nfsbc1v 3066	. . 3 ⊢ Ⅎx[̣y / x]̣φ
6		sbceq1a 3057	. . 3 ⊢ (x = y → (φ ↔ [̣y / x]̣φ))
7	2, 3, 4, 5, 6	cbvrab 2858	. 2 ⊢ {x ∈ B ∣ φ} = {y ∈ B ∣ [̣y / x]̣φ}
8	1, 7	elrab2 2997	1 ⊢ (A ∈ {x ∈ B ∣ φ} ↔ (A ∈ B ∧ [̣A / x]̣φ))

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  Ⅎwnfc 2477  {crab 2619  [̣wsbc 3047

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rab 2624  df-v 2862  df-sbc 3048

This theorem is used by: (None)