Theorem elin2d 3419

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 3416	. 2 |- ( X e. ( A i^i B ) -> X e. B )
3	1, 2	syl 14	1 |- ( ph -> X e. B )

This proof depends on syntax axioms:   -> wi 4   e. wcel 2209   i^i cin 3219

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226

This theorem is used by:  elfi2  7306  fiuni  7312  fifo  7314  sshashneg  11281  hashfibclem  11282  explecnv  12272  bitsinv1  12729  ballotfilemfmpn  13234  nninfdclemp1  13341  idomdomd  14586  sralmod  14787  2idlridld  14844  restbasg  15269  txcnp  15372  blin2  15533  bj-charfun  16833  bj-charfundc  16834