Theorem abss 3266

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 2327	. . 3 |- { x | x e. A } = A
2	1	sseq2i 3224	. 2 |- ( { x | ph } C_ { x | x e. A } <-> { x | ph } C_ A )
3		ss2ab 3265	. 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 1371   e. wcel 2177   {cab 2192   C_ wss 3170

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-in 3176  df-ss 3183

This theorem is referenced by:  abssdv  3271  rabss  3274  uniiunlem  3286  iunss  3977  reliun  4809  funimaexglem  5371