Theorem ss2ab 3078

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 2227	. . 3 |- F/_ x { x | ph }
2		nfab1 2227	. . 3 |- F/_ x { x | ps }
3	1, 2	dfss2f 3005	. 2 |- ( { x | ph } C_ { x | ps } <-> A. x ( x e. { x | ph } -> x e. { x | ps } ) )
4		abid 2073	. . . 4 |- ( x e. { x | ph } <-> ph )
5		abid 2073	. . . 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 1402	. 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 1285   e. wcel 1436   {cab 2071   C_ wss 2988

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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-in 2994  df-ss 3001

This theorem is referenced by:  abss  3079  ssab  3080  ss2abi  3082  ss2abdv  3083  ss2rab  3086  rabss2  3093  iotanul  4961  iotass  4963