Theorem elin2d 3397

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 3394	. 2 ⊢ (𝑋 ∈ (𝐴 ∩ 𝐵) → 𝑋 ∈ 𝐵)
3	1, 2	syl 14	1 ⊢ (𝜑 → 𝑋 ∈ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2202   ∩ cin 3199

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206

This theorem is referenced by:  elfi2  7170  fiuni  7176  fifo  7178  explecnv  12065  bitsinv1  12522  nninfdclemp1  13070  idomdomd  14290  sralmod  14463  2idlridld  14520  restbasg  14891  txcnp  14994  blin2  15155  bj-charfun  16402  bj-charfundc  16403