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Theorem elin3 3101
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 (A 𝑋 ↔ (A B A 𝐶 A 𝐷))

Proof of Theorem elin3
StepHypRef Expression
1 elin 3099 . . 3 (A (B𝐶) ↔ (A B A 𝐶))
21anbi1i 434 . 2 ((A (B𝐶) A 𝐷) ↔ ((A B A 𝐶) A 𝐷))
3 elin3.x . . 3 𝑋 = ((B𝐶) ∩ 𝐷)
43elin2 3100 . 2 (A 𝑋 ↔ (A (B𝐶) A 𝐷))
5 df-3an 873 . 2 ((A B A 𝐶 A 𝐷) ↔ ((A B A 𝐶) A 𝐷))
62, 4, 53bitr4i 201 1 (A 𝑋 ↔ (A B A 𝐶 A 𝐷))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   w3a 871   = wceq 1226   wcel 1370  cin 2889
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-in 2897
This theorem is referenced by: (None)
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