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Theorem elin1d 3339
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 3336 . 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 2160    i^i cin 3143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150
This theorem is referenced by:  fiuni  7008  explecnv  11548  nninfdclemcl  12502  nninfdclemp1  12504  2idllidld  13838  qus1  13858  restbasg  14145  txcnp  14248  blin2  14409  bj-charfun  15037
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