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 2902	. . 3 ⊢ Ⅎ𝑦𝐵
4		nfv 1916	. . 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 3466	. 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 395   ∈ wcel 2105  Ⅎwnfc 2882  {crab 3431  [wsbc 3777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-rab 3432  df-v 3475  df-sbc 3778

This theorem is referenced by:  frpoinsg  6344  wfisgOLD  6355  onminesb  7785  tfisg  7847  mpoxopovel  8211  frinsg  9752  ac6num  10480  hashrabsn1  14341  bnj23  34193  bnj1204  34487  rabrenfdioph  42015