Theorem abss 3216

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 2291	. . 3 |- { x | x e. A } = A
2	1	sseq2i 3174	. 2 |- ( { x | ph } C_ { x | x e. A } <-> { x | ph } C_ A )
3		ss2ab 3215	. 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 1346   e. wcel 2141   {cab 2156   C_ wss 3121

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-in 3127  df-ss 3134

This theorem is referenced by:  abssdv  3221  rabss  3224  uniiunlem  3236  iunss  3914  reliun  4732  funimaexglem  5281