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Theorem ssrd 3143
Description: Deduction 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 1509 . 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 3129 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
73, 6sylibr 133 1  |-  ( ph  ->  A  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1340   F/wnf 1447    e. wcel 2135   F/_wnfc 2293    C_ wss 3112
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-in 3118  df-ss 3125
This theorem is referenced by:  eqrd  3156  exmidomni  7098
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