Theorem rabss2 3576

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		pm3.45 831	. . . 4 |- ( ( x e. A -> x e. B ) -> ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
2	1	alimi 1609	. . 3 |- ( A. x ( x e. A -> x e. B ) -> A. x ( ( x e. A /\ ph ) -> ( x e. B /\ ph ) ) )
3		dfss2 3486	. . 3 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
4		ss2ab 3561	. . 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 266	. 2 |- ( A C_ B -> { x | ( x e. A /\ ph ) } C_ { x | ( x e. B /\ ph ) } )
6		df-rab 2816	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
7		df-rab 2816	. 2 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
8	5, 6, 7	3sstr4g 3538	1 |- ( A C_ B -> { x e. A | ph } C_ { x e. B | ph } )

Syntax hints:   -> wi 4   /\ wa 369   A.wal 1372   e. wcel 1762   {cab 2445   {crab 2811   C_ wss 3469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-rab 2816  df-in 3476  df-ss 3483

This theorem is referenced by:  sess2  4841  suppfnss  6915  hashbcss  14370  dprdss  16859  minveclem4  21575  prmdvdsfi  23102  mumul  23176  sqff1o  23177  rpvmasumlem  23393  disjxwwlkn  24407  clwwlknfi  24440  shatomistici  26942  xpinpreima2  27511  ballotth  28102  rmxyelqirr  30437  idomodle  30747  bj-unrab  33450  lssats  33684  lpssat  33685  lssatle  33687  lssat  33688  atlatmstc  33991  dochspss  36050