Theorem elin2d 3371

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
elin2d	|- ( ph -> X e. B )

Proof of Theorem elin2d

Step	Hyp	Ref	Expression
1		elin1d.1	. 2 |- ( ph -> X e. ( A i^i B ) )
2		elinel2 3368	. 2 |- ( X e. ( A i^i B ) -> X e. B )
3	1, 2	syl 14	1 |- ( ph -> X e. B )

Syntax hints:   -> wi 4   e. wcel 2178   i^i cin 3173

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180

This theorem is referenced by:  elfi2  7100  fiuni  7106  fifo  7108  explecnv  11931  bitsinv1  12388  nninfdclemp1  12936  idomdomd  14154  sralmod  14327  2idlridld  14384  restbasg  14755  txcnp  14858  blin2  15019  bj-charfun  15942  bj-charfundc  15943