Theorem rabss2 3175

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		pm3.45 586	. . . 4 |- ( ( x e. A -> x e. B ) -> ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
2	1	alimi 1431	. . 3 |- ( A. x ( x e. A -> x e. B ) -> A. x ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
3		dfss2 3081	. . 3 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
4		ss2ab 3160	. . 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 200	. 2 |- ( A C_ B -> { x | ( x e. A /\ ph ) } C_ { x | ( x e. B /\ ph ) } )
6		df-rab 2423	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
7		df-rab 2423	. 2 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
8	5, 6, 7	3sstr4g 3135	1 |- ( A C_ B -> { x e. A | ph } C_ { x e. B | ph } )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1329   e. wcel 1480   {cab 2123   {crab 2418   C_ wss 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rab 2423  df-in 3072  df-ss 3079

This theorem is referenced by:  sess2  4255  dvfgg  12815