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Theorem elin2d 3266
Description: Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)
Hypothesis
Ref Expression
elin1d.1  |-  ( ph  ->  X  e.  ( A  i^i  B ) )
Assertion
Ref Expression
elin2d  |-  ( ph  ->  X  e.  B )

Proof of Theorem elin2d
StepHypRef Expression
1 elin1d.1 . 2  |-  ( ph  ->  X  e.  ( A  i^i  B ) )
2 elinel2 3263 . 2  |-  ( X  e.  ( A  i^i  B )  ->  X  e.  B )
31, 2syl 14 1  |-  ( ph  ->  X  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1480    i^i cin 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077
This theorem is referenced by:  elfi2  6860  fiuni  6866  fifo  6868  explecnv  11286  restbasg  12351  txcnp  12454  blin2  12615
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