Theorem ssbrd 4681

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 3273	. 2 ⊢ (φ → (⟨C, D⟩ ∈ A → ⟨C, D⟩ ∈ B))
3		df-br 4641	. 2 ⊢ (CAD ↔ ⟨C, D⟩ ∈ A)
4		df-br 4641	. 2 ⊢ (CBD ↔ ⟨C, D⟩ ∈ B)
5	2, 3, 4	3imtr4g 261	1 ⊢ (φ → (CAD → CBD))

Syntax hints:   → wi 4   ∈ wcel 1710   ⊆ wss 3258  ⟨cop 4562   class class class wbr 4640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-br 4641

This theorem is referenced by:  ssbri  4682  coss1  4873  coss2  4874