Theorem ssbrd 3796

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 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)
5	2, 3, 4	3imtr4g 194	1 ⊢ (φ → (𝐶A𝐷 → 𝐶B𝐷))

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