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Theorem bnj1361 28606
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1361.1  |-  ( ph  ->  A. x ( x  e.  A  ->  x  e.  B ) )
Assertion
Ref Expression
bnj1361  |-  ( ph  ->  A  C_  B )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem bnj1361
StepHypRef Expression
1 bnj1361.1 . 2  |-  ( ph  ->  A. x ( x  e.  A  ->  x  e.  B ) )
2 dfss2 3245 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
31, 2sylibr 203 1  |-  ( ph  ->  A  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1540    e. wcel 1710    C_ wss 3228
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-in 3235  df-ss 3242
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