Theorem elind 3392

Description: Deduce membership in an intersection of two classes. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
elind.1	⊢ (𝜑 → 𝑋 ∈ 𝐴)
elind.2	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
elind	⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵))

Proof of Theorem elind

Step	Hyp	Ref	Expression
1		elind.1	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
2		elind.2	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3		elin 3390	. 2 ⊢ (𝑋 ∈ (𝐴 ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵))
4	1, 2, 3	sylanbrc 417	1 ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 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:  fnfvimad  5889  elfir  7171  infpwfidom  7408  nninfdclemcl  13068  nninfdclemp1  13070  strslfv2d  13124  bassetsnn  13138  insubm  13567  2idl0  14525  2idl1  14526  baspartn  14773  bastg  14784  isopn3  14848  restbasg  14891  lmss  14969  metrest  15229  tgioo  15277  dvmulxxbr  15425  elply2  15458  pilem3  15506  2sqlem7  15849