Theorem ss2abdv 3266

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 1897	. 2 |- ( ph -> A. x ( ps -> ch ) )
3		ss2ab 3261	. 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 1371   {cab 2191   C_ wss 3166

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-in 3172  df-ss 3179

This theorem is referenced by:  ssopab2  4322  iotass  5249  imadif  5354  imain  5356  opabbrex  5989  ssoprab2  6001  tfr1onlemssrecs  6425  tfrcllemssrecs  6438  ss2ixp  6798  ptex  13096  plyss  15210