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Theorem bnj1241 29156
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1241.1  |-  ( ph  ->  A  C_  B )
bnj1241.2  |-  ( ps 
->  C  =  A
)
Assertion
Ref Expression
bnj1241  |-  ( (
ph  /\  ps )  ->  C  C_  B )

Proof of Theorem bnj1241
StepHypRef Expression
1 bnj1241.2 . . . 4  |-  ( ps 
->  C  =  A
)
21eqcomd 2301 . . 3  |-  ( ps 
->  A  =  C
)
32adantl 452 . 2  |-  ( (
ph  /\  ps )  ->  A  =  C )
4 bnj1241.1 . . 3  |-  ( ph  ->  A  C_  B )
54adantr 451 . 2  |-  ( (
ph  /\  ps )  ->  A  C_  B )
63, 5eqsstr3d 3226 1  |-  ( (
ph  /\  ps )  ->  C  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    C_ wss 3165
This theorem is referenced by:  bnj1245  29360  bnj1311  29370
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-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-in 3172  df-ss 3179
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