Theorem ss2rab 3343

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 2624	. . 3 |- {x e. A | ph} = {x | (x e. A /\ ph)}
2		df-rab 2624	. . 3 |- {x e. A | ps} = {x | (x e. A /\ ps)}
3	1, 2	sseq12i 3298	. 2 |- ({x e. A | ph} C_ {x e. A | ps} <-> {x | (x e. A /\ ph)} C_ {x | (x e. A /\ ps)})
4		ss2ab 3335	. 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 2620	. . 3 |- (A.x e. A (ph -> ps) <-> A.x(x e. A -> (ph -> ps)))
6		imdistan 670	. . . 4 |- ((x e. A -> (ph -> ps)) <-> ((x e. A /\ ph) -> (x e. A /\ ps)))
7	6	albii 1566	. . 3 |- (A.x(x e. A -> (ph -> ps)) <-> A.x((x e. A /\ ph) -> (x e. A /\ ps)))
8	5, 7	bitr2i 241	. 2 |- (A.x((x e. A /\ ph) -> (x e. A /\ ps)) <-> A.x e. A (ph -> ps))
9	3, 4, 8	3bitri 262	1 |- ({x e. A | ph} C_ {x e. A | ps} <-> A.x e. A (ph -> ps))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   e. wcel 1710  {cab 2339  A.wral 2615  {crab 2619   C_ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  ss2rabdv  3348  ss2rabi  3349