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Theorem elin2d 3234
 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 3231 . 2 (𝑋 ∈ (𝐴𝐵) → 𝑋𝐵)
31, 2syl 14 1 (𝜑𝑋𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1463   ∩ cin 3038 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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045 This theorem is referenced by:  elfi2  6826  fiuni  6832  fifo  6834  explecnv  11214  restbasg  12232  txcnp  12335  blin2  12496
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