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Theorem ssbrd 3796
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 2938 . 2 (φ → (⟨𝐶, 𝐷 A → ⟨𝐶, 𝐷 B))
3 df-br 3756 . 2 (𝐶A𝐷 ↔ ⟨𝐶, 𝐷 A)
4 df-br 3756 . 2 (𝐶B𝐷 ↔ ⟨𝐶, 𝐷 B)
52, 3, 43imtr4g 194 1 (φ → (𝐶A𝐷𝐶B𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1390  wss 2911  cop 3370   class class class wbr 3755
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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925  df-br 3756
This theorem is referenced by:  ssbri  3797  sess1  4059  brrelex12  4324  coss1  4434  coss2  4435  eqbrrdva  4448  ersym  6054  ertr  6057
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