Theorem elssabg 3923

Description: Membership in a class abstraction involving a subset. Unlike elabg 2739, A does not have to be a set. (Contributed by NM, 29-Aug-2006.)

Hypothesis

Ref	Expression
elssabg.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
elssabg	|- ( B e. V -> ( A e. { x | ( x C_ B /\ ph ) } <-> ( A C_ B /\ ps ) ) )

Distinct variable groups:   x, A   x, B   ps, x

Allowed substitution hints:   ph( x)   V( x)

Proof of Theorem elssabg

Step	Hyp	Ref	Expression
1		ssexg 3917	. . . 4 |- ( ( A C_ B /\ B e. V ) -> A e. _V )
2	1	expcom 114	. . 3 |- ( B e. V -> ( A C_ B -> A e. _V ) )
3	2	adantrd 273	. 2 |- ( B e. V -> ( ( A C_ B /\ ps ) -> A e. _V ) )
4		sseq1 3020	. . . 4 |- ( x = A -> ( x C_ B <-> A C_ B ) )
5		elssabg.1	. . . 4 |- ( x = A -> ( ph <-> ps ) )
6	4, 5	anbi12d 456	. . 3 |- ( x = A -> ( ( x C_ B /\ ph ) <-> ( A C_ B /\ ps ) ) )
7	6	elab3g 2744	. 2 |- ( ( ( A C_ B /\ ps ) -> A e. _V ) -> ( A e. { x | ( x C_ B /\ ph ) } <-> ( A C_ B /\ ps ) ) )
8	3, 7	syl 14	1 |- ( B e. V -> ( A e. { x | ( x C_ B /\ ph ) } <-> ( A C_ B /\ ps ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   {cab 2067   _Vcvv 2601   C_ wss 2973

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  ax-sep 3896

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-v 2603  df-in 2979  df-ss 2986

This theorem is referenced by: (None)