Theorem ss2ab 3215

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 2314	. . 3 |- F/_ x { x | ph }
2		nfab1 2314	. . 3 |- F/_ x { x | ps }
3	1, 2	dfss2f 3138	. 2 |- ( { x | ph } C_ { x | ps } <-> A. x ( x e. { x | ph } -> x e. { x | ps } ) )
4		abid 2158	. . . 4 |- ( x e. { x | ph } <-> ph )
5		abid 2158	. . . 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 1463	. 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 1346   e. wcel 2141   {cab 2156   C_ wss 3121

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-in 3127  df-ss 3134

This theorem is referenced by:  abss  3216  ssab  3217  ss2abi  3219  ss2abdv  3220  ss2rab  3223  rabss2  3230  iotanul  5175  iotass  5177