Theorem ss2abdv 3215

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 1862	. 2 |- ( ph -> A. x ( ps -> ch ) )
3		ss2ab 3210	. 2 |- ( { x | ps } C_ { x | ch } <-> A. x ( ps -> ch ) )
4	2, 3	sylibr 133	1 |- ( ph -> { x | ps } C_ { x | ch } )

Syntax hints:   -> wi 4   A.wal 1341   {cab 2151   C_ wss 3116

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-in 3122  df-ss 3129

This theorem is referenced by:  ssopab2  4253  iotass  5170  imadif  5268  imain  5270  opabbrex  5886  ssoprab2  5898  tfr1onlemssrecs  6307  tfrcllemssrecs  6320  ss2ixp  6677