Theorem elin2d 3174

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 3171	. 2 ⊢ (𝑋 ∈ (𝐴 ∩ 𝐵) → 𝑋 ∈ 𝐵)
3	1, 2	syl 14	1 ⊢ (𝜑 → 𝑋 ∈ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 1434   ∩ cin 2983

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-in 2990

This theorem is referenced by: (None)