Theorem ssbrd 4696

Description: Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)

Hypothesis

Ref	Expression
ssbrd.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
ssbrd	⊢ (𝜑 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))

Proof of Theorem ssbrd

Step	Hyp	Ref	Expression
1		ssbrd.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2	1	sseld 3602	. 2 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∈ 𝐴 → ⟨𝐶, 𝐷⟩ ∈ 𝐵))
3		df-br 4654	. 2 ⊢ (𝐶𝐴𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝐴)
4		df-br 4654	. 2 ⊢ (𝐶𝐵𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝐵)
5	2, 3, 4	3imtr4g 285	1 ⊢ (𝜑 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))

Syntax hints:   → wi 4   ∈ wcel 1990   ⊆ wss 3574  ⟨cop 4183   class class class wbr 4653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588  df-br 4654

This theorem is referenced by:  ssbri  4697  sess1  5082  brrelex12  5155  coss1  5277  coss2  5278  eqbrrdva  5291  cnvss  5294  ssrelrn  5315  ersym  7754  ertr  7757  fpwwe2lem6  9457  fpwwe2lem7  9458  fpwwe2lem9  9460  fpwwe2lem12  9463  fpwwe2lem13  9464  fpwwe2  9465  coss12d  13711  fthres2  16592  invfuc  16634  pospo  16973  dirref  17235  efgcpbl  18169  frgpuplem  18185  subrguss  18795  znleval  19903  ustref  22022  ustuqtop4  22048  isucn2  22083  brelg  29421  metider  29937  mclsppslem  31480  fundmpss  31664  iunrelexpuztr  38011  frege96d  38041  frege91d  38043  frege98d  38045  frege124d  38053