Theorem ss2ab 3160

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 2281	. . 3 |- F/_ x { x | ph }
2		nfab1 2281	. . 3 |- F/_ x { x | ps }
3	1, 2	dfss2f 3083	. 2 |- ( { x | ph } C_ { x | ps } <-> A. x ( x e. { x | ph } -> x e. { x | ps } ) )
4		abid 2125	. . . 4 |- ( x e. { x | ph } <-> ph )
5		abid 2125	. . . 4 |- ( x e. { x | ps } <-> ps )
6	4, 5	imbi12i 238	. . 3 |- ( ( x e. { x | ph } -> x e. { x | ps } ) <-> ( ph -> ps ) )
7	6	albii 1446	. 2 |- ( A. x ( x e. { x | ph } -> x e. { x | ps } ) <-> A. x ( ph -> ps ) )
8	3, 7	bitri 183	1 |- ( { x | ph } C_ { x | ps } <-> A. x ( ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   e. wcel 1480   {cab 2123   C_ wss 3066

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-in 3072  df-ss 3079

This theorem is referenced by:  abss  3161  ssab  3162  ss2abi  3164  ss2abdv  3165  ss2rab  3168  rabss2  3175  iotanul  5098  iotass  5100