Theorem abss 3671

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 2745	. . 3 |- { x | x e. A } = A
2	1	sseq2i 3630	. 2 |- ( { x | ph } C_ { x | x e. A } <-> { x | ph } C_ A )
3		ss2ab 3670	. 2 |- ( { x | ph } C_ { x | x e. A } <-> A. x ( ph -> x e. A ) )
4	2, 3	bitr3i 266	1 |- ( { x | ph } C_ A <-> A. x ( ph -> x e. A ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   e. wcel 1990   {cab 2608   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-in 3581  df-ss 3588

This theorem is referenced by:  abssdv  3676  rabss  3679  uniiunlem  3691  iunss  4561  moabex  4927  reliun  5239  axdc2lem  9270  mptelee  25775  fpwrelmap  29508  ss2iundf  37951  iunssf  39263  hoidmvlelem1  40809