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Theorem elin2d 3297
Description: Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)
Hypothesis
Ref Expression
elin1d.1 (𝜑𝑋 ∈ (𝐴𝐵))
Assertion
Ref Expression
elin2d (𝜑𝑋𝐵)

Proof of Theorem elin2d
StepHypRef Expression
1 elin1d.1 . 2 (𝜑𝑋 ∈ (𝐴𝐵))
2 elinel2 3294 . 2 (𝑋 ∈ (𝐴𝐵) → 𝑋𝐵)
31, 2syl 14 1 (𝜑𝑋𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2128  cin 3101
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108
This theorem is referenced by:  elfi2  6917  fiuni  6923  fifo  6925  explecnv  11406  nninfdclemp1  12223  restbasg  12610  txcnp  12713  blin2  12874  bj-charfun  13424  bj-charfundc  13425
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