Theorem rabss2 3077

Description: Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
rabss2	|- ( A C_ B -> { x e. A | ph } C_ { x e. B | ph } )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem rabss2

Step	Hyp	Ref	Expression
1		pm3.45 561	. . . 4 |- ( ( x e. A -> x e. B ) -> ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
2	1	alimi 1384	. . 3 |- ( A. x ( x e. A -> x e. B ) -> A. x ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
3		dfss2 2988	. . 3 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
4		ss2ab 3062	. . 3 |- ( { x | ( x e. A /\ ph ) } C_ { x | ( x e. B /\ ph ) } <-> A. x ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
5	2, 3, 4	3imtr4i 199	. 2 |- ( A C_ B -> { x | ( x e. A /\ ph ) } C_ { x | ( x e. B /\ ph ) } )
6		df-rab 2357	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
7		df-rab 2357	. 2 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
8	5, 6, 7	3sstr4g 3040	1 |- ( A C_ B -> { x e. A | ph } C_ { x e. B | ph } )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   e. wcel 1433   {cab 2067   {crab 2352   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

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-in 2979  df-ss 2986

This theorem is referenced by:  sess2  4093