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Theorem elin3 4174
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 4166 . . 3 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
21anbi1i 623 . 2 ((𝐴 ∈ (𝐵𝐶) ∧ 𝐴𝐷) ↔ ((𝐴𝐵𝐴𝐶) ∧ 𝐴𝐷))
3 elin3.x . . 3 𝑋 = ((𝐵𝐶) ∩ 𝐷)
43elin2 4171 . 2 (𝐴𝑋 ↔ (𝐴 ∈ (𝐵𝐶) ∧ 𝐴𝐷))
5 df-3an 1081 . 2 ((𝐴𝐵𝐴𝐶𝐴𝐷) ↔ ((𝐴𝐵𝐴𝐶) ∧ 𝐴𝐷))
62, 4, 53bitr4i 304 1 (𝐴𝑋 ↔ (𝐴𝐵𝐴𝐶𝐴𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  cin 3932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-in 3940
This theorem is referenced by: (None)
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