Theorem ss2ab 3073

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 2225	. . 3 |- F/_ x { x | ph }
2		nfab1 2225	. . 3 |- F/_ x { x | ps }
3	1, 2	dfss2f 3001	. 2 |- ( { x | ph } C_ { x | ps } <-> A. x ( x e. { x | ph } -> x e. { x | ps } ) )
4		abid 2071	. . . 4 |- ( x e. { x | ph } <-> ph )
5		abid 2071	. . . 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 1400	. 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 1283   e. wcel 1434   {cab 2069   C_ wss 2984

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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-in 2990  df-ss 2997

This theorem is referenced by:  abss  3074  ssab  3075  ss2abi  3077  ss2abdv  3078  ss2rab  3081  rabss2  3088  iotanul  4947  iotass  4949