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Theorem elin1d 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
elin1d  |-  ( ph  ->  X  e.  A )

Proof of Theorem elin1d
StepHypRef Expression
1 elin1d.1 . 2  |-  ( ph  ->  X  e.  ( A  i^i  B ) )
2 elinel1 3263 . 2  |-  ( X  e.  ( A  i^i  B )  ->  X  e.  A )
31, 2syl 14 1  |-  ( ph  ->  X  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1481    i^i cin 3071
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2689  df-in 3078
This theorem is referenced by:  fiuni  6870  explecnv  11302  restbasg  12367  txcnp  12470  blin2  12631
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