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Theorem bnj1361 29274
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 3339 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
31, 2sylibr 205 1  |-  ( ph  ->  A  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550    e. wcel 1726    C_ wss 3322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-in 3329  df-ss 3336
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