Theorem elrabsf 2852

Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2747 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	|- F/_ x B

Assertion

Ref	Expression
elrabsf	|- ( A e. { x e. B | ph } <-> ( A e. B /\ [. A / x ]. ph ) )

Proof of Theorem elrabsf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq 2817	. 2 |- ( y = A -> ( [. y / x ]. ph <-> [. A / x ]. ph ) )
2		elrabsf.1	. . 3 |- F/_ x B
3		nfcv 2219	. . 3 |- F/_ y B
4		nfv 1461	. . 3 |- F/ y ph
5		nfsbc1v 2833	. . 3 |- F/ x [. y / x ]. ph
6		sbceq1a 2824	. . 3 |- ( x = y -> ( ph <-> [. y / x ]. ph ) )
7	2, 3, 4, 5, 6	cbvrab 2599	. 2 |- { x e. B | ph } = { y e. B | [. y / x ]. ph }
8	1, 7	elrab2 2751	1 |- ( A e. { x e. B | ph } <-> ( A e. B /\ [. A / x ]. ph ) )

Syntax hints:   /\ wa 102   <-> wb 103   e. wcel 1433   F/_wnfc 2206   {crab 2352   [.wsbc 2815

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-v 2603  df-sbc 2816

This theorem is referenced by:  mpt2xopovel  5879  zsupcllemstep  10341  infssuzex  10345