Theorem ss2abdv 3007

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 1751	. 2 |- (ph -> A.x(ps -> ch))
3		ss2ab 3002	. 2 |- ( {x | ps } C_ {x | ch } <-> A.x(ps -> ch))
4	2, 3	sylibr 137	1 |- (ph -> {x | ps } C_ {x | ch })

Syntax hints:   -> wi 4  A.wal 1240   {cab 2023   C_ wss 2911

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925

This theorem is referenced by:  ssopab2  4003  iotass  4827  imadif  4922  imain  4924  opabbrex  5491  ssoprab2  5503