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Theorem elin1d 3204
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 3201 . 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 1445    i^i cin 3012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-in 3019
This theorem is referenced by:  explecnv  11048  restbasg  12020  blin2  12218
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