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Theorem elin3 4130
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 3900 . . 3 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
21anbi1i 626 . 2 ((𝐴 ∈ (𝐵𝐶) ∧ 𝐴𝐷) ↔ ((𝐴𝐵𝐴𝐶) ∧ 𝐴𝐷))
3 elin3.x . . 3 𝑋 = ((𝐵𝐶) ∩ 𝐷)
43elin2 4127 . 2 (𝐴𝑋 ↔ (𝐴 ∈ (𝐵𝐶) ∧ 𝐴𝐷))
5 df-3an 1086 . 2 ((𝐴𝐵𝐴𝐶𝐴𝐷) ↔ ((𝐴𝐵𝐴𝐶) ∧ 𝐴𝐷))
62, 4, 53bitr4i 306 1 (𝐴𝑋 ↔ (𝐴𝐵𝐴𝐶𝐴𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  cin 3883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-in 3891
This theorem is referenced by: (None)
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