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Theorem elinti 3624
Description: Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
elinti  |-  ( A  e.  |^| B  ->  ( C  e.  B  ->  A  e.  C ) )

Proof of Theorem elinti
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elintg 3623 . . 3  |-  ( A  e.  |^| B  ->  ( A  e.  |^| B  <->  A. x  e.  B  A  e.  x ) )
2 eleq2 2101 . . . 4  |-  ( x  =  C  ->  ( A  e.  x  <->  A  e.  C ) )
32rspccv 2653 . . 3  |-  ( A. x  e.  B  A  e.  x  ->  ( C  e.  B  ->  A  e.  C ) )
41, 3syl6bi 152 . 2  |-  ( A  e.  |^| B  ->  ( A  e.  |^| B  -> 
( C  e.  B  ->  A  e.  C ) ) )
54pm2.43i 43 1  |-  ( A  e.  |^| B  ->  ( C  e.  B  ->  A  e.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1393   A.wral 2306   |^|cint 3615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-int 3616
This theorem is referenced by: (None)
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