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Theorem ssrd 23141
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 1757 . 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 3184 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
73, 6sylibr 203 1  |-  ( ph  ->  A  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530   F/wnf 1534    e. wcel 1696   F/_wnfc 2419    C_ wss 3165
This theorem is referenced by:  eqrd  23142  rabss3d  23152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-in 3172  df-ss 3179
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