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Theorem ssrd 3353
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 1781 . 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 3339 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
73, 6sylibr 204 1  |-  ( ph  ->  A  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549   F/wnf 1553    e. wcel 1725   F/_wnfc 2559    C_ wss 3320
This theorem is referenced by:  eqrd  3366  neiptopnei  17196  rabss3d  23995
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-in 3327  df-ss 3334
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