Theorem rabss2 3685

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		pm3.45 879	. . . 4 |- ( ( x e. A -> x e. B ) -> ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
2	1	alimi 1739	. . 3 |- ( A. x ( x e. A -> x e. B ) -> A. x ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
3		dfss2 3591	. . 3 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
4		ss2ab 3670	. . 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 281	. 2 |- ( A C_ B -> { x | ( x e. A /\ ph ) } C_ { x | ( x e. B /\ ph ) } )
6		df-rab 2921	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
7		df-rab 2921	. 2 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
8	5, 6, 7	3sstr4g 3646	1 |- ( A C_ B -> { x e. A | ph } C_ { x e. B | ph } )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   e. wcel 1990   {cab 2608   {crab 2916   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-in 3581  df-ss 3588

This theorem is referenced by:  sess2  5083  suppfnss  7320  hashbcss  15708  dprdss  18428  minveclem4  23203  prmdvdsfi  24833  mumul  24907  sqff1o  24908  rpvmasumlem  25176  disjxwwlkn  26808  clwwlksnfi  26913  shatomistici  29220  rabfodom  29344  xpinpreima2  29953  ballotth  30599  bj-unrab  32922  icorempt2  33199  lssats  34299  lpssat  34300  lssatle  34302  lssat  34303  atlatmstc  34606  dochspss  36667  rmxyelqirr  37475  idomodle  37774  sssmf  40947