Theorem elrabsf 3003

Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2893 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 2966	. 2 |- ( y = A -> ( [. y / x ]. ph <-> [. A / x ]. ph ) )
2		elrabsf.1	. . 3 |- F/_ x B
3		nfcv 2319	. . 3 |- F/_ y B
4		nfv 1528	. . 3 |- F/ y ph
5		nfsbc1v 2983	. . 3 |- F/ x [. y / x ]. ph
6		sbceq1a 2974	. . 3 |- ( x = y -> ( ph <-> [. y / x ]. ph ) )
7	2, 3, 4, 5, 6	cbvrab 2737	. 2 |- { x e. B | ph } = { y e. B | [. y / x ]. ph }
8	1, 7	elrab2 2898	1 |- ( A e. { x e. B | ph } <-> ( A e. B /\ [. A / x ]. ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2148   F/_wnfc 2306   {crab 2459   [.wsbc 2964

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2741  df-sbc 2965

This theorem is referenced by:  mpoxopovel  6244  zsupcllemstep  11948  infssuzex  11952