Theorem ss2abdv 3220

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 1867	. 2 |- ( ph -> A. x ( ps -> ch ) )
3		ss2ab 3215	. 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 1346   {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:  ssopab2  4260  iotass  5177  imadif  5278  imain  5280  opabbrex  5897  ssoprab2  5909  tfr1onlemssrecs  6318  tfrcllemssrecs  6331  ss2ixp  6689