Theorem elin2d 3803

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 3800	. 2 |- ( X e. ( A i^i B ) -> X e. B )
3	1, 2	syl 17	1 |- ( ph -> X e. B )

Syntax hints:   -> wi 4   e. wcel 1990   i^i cin 3573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581

This theorem is referenced by:  bitsinv1  15164  txkgen  21455  nmoleub2lem3  22915  nmoleub3  22919  tayl0  24116  esum2d  30155  ispisys2  30216  sigapisys  30218  sigapildsyslem  30224  sigapildsys  30225  tgoldbachgt  30741  bnj1172  31069  cnrefiisplem  40055  hoiqssbllem3  40838  smflimlem3  40981