Theorem abss 3293

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 2350	. . 3 |- { x | x e. A } = A
2	1	sseq2i 3251	. 2 |- ( { x | ph } C_ { x | x e. A } <-> { x | ph } C_ A )
3		ss2ab 3292	. 2 |- ( { x | ph } C_ { x | x e. A } <-> A. x ( ph -> x e. A ) )
4	2, 3	bitr3i 186	1 |- ( { x | ph } C_ A <-> A. x ( ph -> x e. A ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1393   e. wcel 2200   {cab 2215   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-in 3203  df-ss 3210

This theorem is referenced by:  abssdv  3298  rabss  3301  uniiunlem  3313  iunss  4005  reliun  4839  funimaexglem  5403