Theorem elin2d 3337

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 3334	. 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 2158   i^i cin 3140

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-in 3147

This theorem is referenced by:  elfi2  6984  fiuni  6990  fifo  6992  explecnv  11526  nninfdclemp1  12464  sralmod  13634  2idlridld  13686  restbasg  13939  txcnp  14042  blin2  14203  bj-charfun  14830  bj-charfundc  14831