Theorem ssbrd 3878

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 3022	. 2 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∈ 𝐴 → ⟨𝐶, 𝐷⟩ ∈ 𝐵))
3		df-br 3838	. 2 ⊢ (𝐶𝐴𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝐴)
4		df-br 3838	. 2 ⊢ (𝐶𝐵𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝐵)
5	2, 3, 4	3imtr4g 203	1 ⊢ (𝜑 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))

Syntax hints:   → wi 4   ∈ wcel 1438   ⊆ wss 2997  ⟨cop 3444   class class class wbr 3837

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3003  df-ss 3010  df-br 3838

This theorem is referenced by:  ssbri  3879  sess1  4155  brrelex12  4464  coss1  4579  coss2  4580  eqbrrdva  4594  ersym  6284  ertr  6287