Theorem elin2d 3363

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 3360	. 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 2176   i^i cin 3165

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172

This theorem is referenced by:  elfi2  7074  fiuni  7080  fifo  7082  explecnv  11816  bitsinv1  12273  nninfdclemp1  12821  idomdomd  14039  sralmod  14212  2idlridld  14269  restbasg  14640  txcnp  14743  blin2  14904  bj-charfun  15743  bj-charfundc  15744