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Theorem bnj1241 28517
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 2392 . . 3  |-  ( ps 
->  A  =  C
)
32adantl 453 . 2  |-  ( (
ph  /\  ps )  ->  A  =  C )
4 bnj1241.1 . . 3  |-  ( ph  ->  A  C_  B )
54adantr 452 . 2  |-  ( (
ph  /\  ps )  ->  A  C_  B )
63, 5eqsstr3d 3326 1  |-  ( (
ph  /\  ps )  ->  C  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    C_ wss 3263
This theorem is referenced by:  bnj1245  28721  bnj1311  28731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-in 3270  df-ss 3277
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