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Theorem elin3 3787
Description: Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Hypothesis
Ref Expression
elin3.x 𝑋 = ((𝐵𝐶) ∩ 𝐷)
Assertion
Ref Expression
elin3 (𝐴𝑋 ↔ (𝐴𝐵𝐴𝐶𝐴𝐷))

Proof of Theorem elin3
StepHypRef Expression
1 elin 3779 . . 3 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
21anbi1i 730 . 2 ((𝐴 ∈ (𝐵𝐶) ∧ 𝐴𝐷) ↔ ((𝐴𝐵𝐴𝐶) ∧ 𝐴𝐷))
3 elin3.x . . 3 𝑋 = ((𝐵𝐶) ∩ 𝐷)
43elin2 3784 . 2 (𝐴𝑋 ↔ (𝐴 ∈ (𝐵𝐶) ∧ 𝐴𝐷))
5 df-3an 1038 . 2 ((𝐴𝐵𝐴𝐶𝐴𝐷) ↔ ((𝐴𝐵𝐴𝐶) ∧ 𝐴𝐷))
62, 4, 53bitr4i 292 1 (𝐴𝑋 ↔ (𝐴𝐵𝐴𝐶𝐴𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  cin 3558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-in 3566
This theorem is referenced by: (None)
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