Theorem ss2ab 3062

Description: Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)

Assertion

Ref	Expression
ss2ab	|- ( { x | ph } C_ { x | ps } <-> A. x ( ph -> ps ) )

Proof of Theorem ss2ab

Step	Hyp	Ref	Expression
1		nfab1 2221	. . 3 |- F/_ x { x | ph }
2		nfab1 2221	. . 3 |- F/_ x { x | ps }
3	1, 2	dfss2f 2990	. 2 |- ( { x | ph } C_ { x | ps } <-> A. x ( x e. { x | ph } -> x e. { x | ps } ) )
4		abid 2069	. . . 4 |- ( x e. { x | ph } <-> ph )
5		abid 2069	. . . 4 |- ( x e. { x | ps } <-> ps )
6	4, 5	imbi12i 237	. . 3 |- ( ( x e. { x | ph } -> x e. { x | ps } ) <-> ( ph -> ps ) )
7	6	albii 1399	. 2 |- ( A. x ( x e. { x | ph } -> x e. { x | ps } ) <-> A. x ( ph -> ps ) )
8	3, 7	bitri 182	1 |- ( { x | ph } C_ { x | ps } <-> A. x ( ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   e. wcel 1433   {cab 2067   C_ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-in 2979  df-ss 2986

This theorem is referenced by:  abss  3063  ssab  3064  ss2abi  3066  ss2abdv  3067  ss2rab  3070  rabss2  3077  iotanul  4902  iotass  4904