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Theorem elin3 3125
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 3123 . . 3 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
21anbi1i 431 . 2 ((𝐴 ∈ (𝐵𝐶) ∧ 𝐴𝐷) ↔ ((𝐴𝐵𝐴𝐶) ∧ 𝐴𝐷))
3 elin3.x . . 3 𝑋 = ((𝐵𝐶) ∩ 𝐷)
43elin2 3124 . 2 (𝐴𝑋 ↔ (𝐴 ∈ (𝐵𝐶) ∧ 𝐴𝐷))
5 df-3an 887 . 2 ((𝐴𝐵𝐴𝐶𝐴𝐷) ↔ ((𝐴𝐵𝐴𝐶) ∧ 𝐴𝐷))
62, 4, 53bitr4i 201 1 (𝐴𝑋 ↔ (𝐴𝐵𝐴𝐶𝐴𝐷))
Colors of variables: wff set class
Syntax hints:  wa 97  wb 98  w3a 885   = wceq 1243  wcel 1393  cin 2913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-in 2921
This theorem is referenced by: (None)
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