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Theorem brin 3781
 Description: The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
Assertion
Ref Expression
brin (A(𝑅𝑆)B ↔ (A𝑅B A𝑆B))

Proof of Theorem brin
StepHypRef Expression
1 elin 3099 . 2 (⟨A, B (𝑅𝑆) ↔ (⟨A, B 𝑅 A, B 𝑆))
2 df-br 3735 . 2 (A(𝑅𝑆)B ↔ ⟨A, B (𝑅𝑆))
3 df-br 3735 . . 3 (A𝑅B ↔ ⟨A, B 𝑅)
4 df-br 3735 . . 3 (A𝑆B ↔ ⟨A, B 𝑆)
53, 4anbi12i 436 . 2 ((A𝑅B A𝑆B) ↔ (⟨A, B 𝑅 A, B 𝑆))
61, 2, 53bitr4i 201 1 (A(𝑅𝑆)B ↔ (A𝑅B A𝑆B))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   ∈ wcel 1370   ∩ cin 2889  ⟨cop 3349   class class class wbr 3734 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-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  df-br 3735 This theorem is referenced by:  brinxp2  4330  trin2  4639  poirr2  4640  cnvin  4654  tpostpos  5797  erinxp  6087
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