Theorem ssbrd 2094

Description: Deduction from a subclass relationship of binary relations.

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 1506	. 2 ⊢ (φ → (⟨C, D⟩ ∈ A → ⟨C, D⟩ ∈ B))
3		df-br 2063	. 2 ⊢ (CAD ↔ ⟨C, D⟩ ∈ A)
4		df-br 2063	. 2 ⊢ (CBD ↔ ⟨C, D⟩ ∈ B)
5	2, 3, 4	3imtr4g 426	1 ⊢ (φ → (CAD → CBD))

Syntax hints:   → wi 2   ∈ wcel 1092   ⊆ wss 1487  ⟨cop 1810   class class class wbr 2054

This theorem is referenced by:  endom 3289

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492  df-br 2063

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