Theorem abss 3171

Description: Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)

Assertion

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

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem abss

Step	Hyp	Ref	Expression
1		abid2 2261	. . 3 |- { x | x e. A } = A
2	1	sseq2i 3129	. 2 |- ( { x | ph } C_ { x | x e. A } <-> { x | ph } C_ A )
3		ss2ab 3170	. 2 |- ( { x | ph } C_ { x | x e. A } <-> A. x ( ph -> x e. A ) )
4	2, 3	bitr3i 185	1 |- ( { x | ph } C_ A <-> A. x ( ph -> x e. A ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1330   e. wcel 1481   {cab 2126   C_ wss 3076

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-in 3082  df-ss 3089

This theorem is referenced by:  abssdv  3176  rabss  3179  uniiunlem  3190  iunss  3862  reliun  4668  funimaexglem  5214