Theorem ss2abdv 3265

Description: Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)

Hypothesis

Ref	Expression
ss2abdv.1	|- ( ph -> ( ps -> ch ) )

Assertion

Ref	Expression
ss2abdv	|- ( ph -> { x | ps } C_ { x | ch } )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   ch( x)

Proof of Theorem ss2abdv

Step	Hyp	Ref	Expression
1		ss2abdv.1	. . 3 |- ( ph -> ( ps -> ch ) )
2	1	alrimiv 1896	. 2 |- ( ph -> A. x ( ps -> ch ) )
3		ss2ab 3260	. 2 |- ( { x | ps } C_ { x | ch } <-> A. x ( ps -> ch ) )
4	2, 3	sylibr 134	1 |- ( ph -> { x | ps } C_ { x | ch } )

Syntax hints:   -> wi 4   A.wal 1370   {cab 2190   C_ wss 3165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-in 3171  df-ss 3178

This theorem is referenced by:  ssopab2  4321  iotass  5248  imadif  5353  imain  5355  opabbrex  5988  ssoprab2  6000  tfr1onlemssrecs  6424  tfrcllemssrecs  6437  ss2ixp  6797  ptex  13067  plyss  15181