Theorem ss2rab 3300

Description: Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)

Assertion

Ref	Expression
ss2rab	|- ( { x e. A | ph } C_ { x e. A | ps } <-> A. x e. A ( ph -> ps ) )

Proof of Theorem ss2rab

Step	Hyp	Ref	Expression
1		df-rab 2517	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		df-rab 2517	. . 3 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
3	1, 2	sseq12i 3252	. 2 |- ( { x e. A | ph } C_ { x e. A | ps } <-> { x | ( x e. A /\ ph ) } C_ { x | ( x e. A /\ ps ) } )
4		ss2ab 3292	. 2 |- ( { x | ( x e. A /\ ph ) } C_ { x | ( x e. A /\ ps ) } <-> A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
5		df-ral 2513	. . 3 |- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
6		imdistan 444	. . . 4 |- ( ( x e. A -> ( ph -> ps ) ) <-> ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
7	6	albii 1516	. . 3 |- ( A. x ( x e. A -> ( ph -> ps ) ) <-> A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
8	5, 7	bitr2i 185	. 2 |- ( A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) <-> A. x e. A ( ph -> ps ) )
9	3, 4, 8	3bitri 206	1 |- ( { x e. A | ph } C_ { x e. A | ps } <-> A. x e. A ( ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1393   e. wcel 2200   {cab 2215   A.wral 2508   {crab 2512   C_ wss 3197

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rab 2517  df-in 3203  df-ss 3210

This theorem is referenced by:  ss2rabdv  3305  ss2rabi  3306