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Theorem ssrd 3017
Description: Deduction rule based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)
Hypotheses
Ref Expression
ssrd.0  |-  F/ x ph
ssrd.1  |-  F/_ x A
ssrd.2  |-  F/_ x B
ssrd.3  |-  ( ph  ->  ( x  e.  A  ->  x  e.  B ) )
Assertion
Ref Expression
ssrd  |-  ( ph  ->  A  C_  B )

Proof of Theorem ssrd
StepHypRef Expression
1 ssrd.0 . . 3  |-  F/ x ph
2 ssrd.3 . . 3  |-  ( ph  ->  ( x  e.  A  ->  x  e.  B ) )
31, 2alrimi 1458 . 2  |-  ( ph  ->  A. x ( x  e.  A  ->  x  e.  B ) )
4 ssrd.1 . . 3  |-  F/_ x A
5 ssrd.2 . . 3  |-  F/_ x B
64, 5dfss2f 3003 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
73, 6sylibr 132 1  |-  ( ph  ->  A  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1285   F/wnf 1392    e. wcel 1436   F/_wnfc 2212    C_ wss 2986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-in 2992  df-ss 2999
This theorem is referenced by:  eqrd  3030  exmidomni  6719
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