Theorem elrabsf 3090

Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2980 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 3053	. 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
2		elrabsf.1	. . 3 ⊢ Ⅎ𝑥𝐵
3		nfcv 2392	. . 3 ⊢ Ⅎ𝑦𝐵
4		nfv 1581	. . 3 ⊢ Ⅎ𝑦𝜑
5		nfsbc1v 3070	. . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
6		sbceq1a 3061	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
7	2, 3, 4, 5, 6	cbvrab 2819	. 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ [𝑦 / 𝑥]𝜑}
8	1, 7	elrab2 2985	1 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∈ wcel 2209  Ⅎwnfc 2379  {crab 2532  [wsbc 3051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-sbc 3052

This theorem is referenced by:  mpoxopovel  6502  zsupcllemstep  10640  infssuzex  10644  infssfzcldc  10647  infssfzledc  10648