Theorem elrabsf 3825

Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3679 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 3779	. 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
2		elrabsf.1	. . 3 ⊢ Ⅎ𝑥𝐵
3		nfcv 2903	. . 3 ⊢ Ⅎ𝑦𝐵
4		nfv 1917	. . 3 ⊢ Ⅎ𝑦𝜑
5		nfsbc1v 3797	. . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
6		sbceq1a 3788	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
7	2, 3, 4, 5, 6	cbvrabw 3467	. 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ [𝑦 / 𝑥]𝜑}
8	1, 7	elrab2 3686	1 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  Ⅎwnfc 2883  {crab 3432  [wsbc 3777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-rab 3433  df-v 3476  df-sbc 3778

This theorem is referenced by:  frpoinsg  6344  wfisgOLD  6355  onminesb  7780  tfisg  7842  mpoxopovel  8204  frinsg  9745  ac6num  10473  hashrabsn1  14333  bnj23  33724  bnj1204  34018  rabrenfdioph  41542