Theorem elssabg 4178

Description: Membership in a class abstraction involving a subset. Unlike elabg 2907, 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 4169	. . . 4 |- ( ( A C_ B /\ B e. V ) -> A e. _V )
2	1	expcom 116	. . 3 |- ( B e. V -> ( A C_ B -> A e. _V ) )
3	2	adantrd 279	. 2 |- ( B e. V -> ( ( A C_ B /\ ps ) -> A e. _V ) )
4		sseq1 3203	. . . 4 |- ( x = A -> ( x C_ B <-> A C_ B ) )
5		elssabg.1	. . . 4 |- ( x = A -> ( ph <-> ps ) )
6	4, 5	anbi12d 473	. . 3 |- ( x = A -> ( ( x C_ B /\ ph ) <-> ( A C_ B /\ ps ) ) )
7	6	elab3g 2912	. 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 104   <-> wb 105   = wceq 1364   e. wcel 2164   {cab 2179   _Vcvv 2760   C_ wss 3154

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-sep 4148

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3160  df-ss 3167

This theorem is referenced by: (None)