Theorem elrabsf 3799

Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3655 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 3755	. 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
2		elrabsf.1	. . 3 ⊢ Ⅎ𝑥𝐵
3		nfcv 2891	. . 3 ⊢ Ⅎ𝑦𝐵
4		nfv 1914	. . 3 ⊢ Ⅎ𝑦𝜑
5		nfsbc1v 3773	. . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
6		sbceq1a 3764	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
7	2, 3, 4, 5, 6	cbvrabw 3441	. 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ [𝑦 / 𝑥]𝜑}
8	1, 7	elrab2 3662	1 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  Ⅎwnfc 2876  {crab 3405  [wsbc 3753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-rab 3406  df-v 3449  df-sbc 3754

This theorem is referenced by:  frpoinsg  6316  onminesb  7769  tfisg  7830  mpoxopovel  8199  frinsg  9704  ac6num  10432  hashrabsn1  14339  bnj23  34708  bnj1204  35002  weiunlem2  36451  rabrenfdioph  42802