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Theorem ssbrd 3775
Description: Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)
Hypothesis
Ref Expression
ssbrd.1 (φAB)
Assertion
Ref Expression
ssbrd (φ → (𝐶A𝐷𝐶B𝐷))

Proof of Theorem ssbrd
StepHypRef Expression
1 ssbrd.1 . . 3 (φAB)
21sseld 2917 . 2 (φ → (⟨𝐶, 𝐷 A → ⟨𝐶, 𝐷 B))
3 df-br 3735 . 2 (𝐶A𝐷 ↔ ⟨𝐶, 𝐷 A)
4 df-br 3735 . 2 (𝐶B𝐷 ↔ ⟨𝐶, 𝐷 B)
52, 3, 43imtr4g 194 1 (φ → (𝐶A𝐷𝐶B𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1370  wss 2890  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-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-11 1374  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-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-in 2897  df-ss 2904  df-br 3735
This theorem is referenced by:  ssbri  3776  sess1  4038  brrelex12  4304  coss1  4414  coss2  4415  eqbrrdva  4428  ersym  6025  ertr  6028
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