Theorem elrabsf 2877

Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2769 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 2842	. 2 |- ( y = A -> ( [. y / x ]. ph <-> [. A / x ]. ph ) )
2		elrabsf.1	. . 3 |- F/_ x B
3		nfcv 2228	. . 3 |- F/_ y B
4		nfv 1466	. . 3 |- F/ y ph
5		nfsbc1v 2858	. . 3 |- F/ x [. y / x ]. ph
6		sbceq1a 2849	. . 3 |- ( x = y -> ( ph <-> [. y / x ]. ph ) )
7	2, 3, 4, 5, 6	cbvrab 2617	. 2 |- { x e. B | ph } = { y e. B | [. y / x ]. ph }
8	1, 7	elrab2 2774	1 |- ( A e. { x e. B | ph } <-> ( A e. B /\ [. A / x ]. ph ) )

Syntax hints:   /\ wa 102   <-> wb 103   e. wcel 1438   F/_wnfc 2215   {crab 2363   [.wsbc 2840

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-sbc 2841

This theorem is referenced by:  mpt2xopovel  6006  zsupcllemstep  11215  infssuzex  11219