Theorem elin2d 3413

Description: Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)

Hypothesis

Ref	Expression
elin1d.1	⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵))

Assertion

Ref	Expression
elin2d	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Proof of Theorem elin2d

Step	Hyp	Ref	Expression
1		elin1d.1	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵))
2		elinel2 3410	. 2 ⊢ (𝑋 ∈ (𝐴 ∩ 𝐵) → 𝑋 ∈ 𝐵)
3	1, 2	syl 14	1 ⊢ (𝜑 → 𝑋 ∈ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2205   ∩ cin 3213

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3220

This theorem is referenced by:  elfi2  7273  fiuni  7279  fifo  7281  sshashneg  11234  hashfibclem  11235  explecnv  12221  bitsinv1  12678  ballotfilemfmpn  13183  nninfdclemp1  13290  idomdomd  14529  sralmod  14729  2idlridld  14786  restbasg  15164  txcnp  15267  blin2  15428  bj-charfun  16718  bj-charfundc  16719