Theorem ssbrd 2656

Description: Deduction from a subclass relationship of binary relations.

Hypothesis

Ref	Expression
ssbrd.1	|- (ph -> A (_ B)

Assertion

Ref	Expression
ssbrd	|- (ph -> (CAD -> CBD))

Proof of Theorem ssbrd

Step	Hyp	Ref	Expression
1		ssbrd.1	. . 3 |- (ph -> A (_ B)
2	1	sseld 2067	. 2 |- (ph -> (<.C, D>. e. A -> <.C, D>. e. B))
3		df-br 2620	. 2 |- (CAD <-> <.C, D>. e. A)
4		df-br 2620	. 2 |- (CBD <-> <.C, D>. e. B)
5	2, 3, 4	3imtr4g 553	1 |- (ph -> (CAD -> CBD))

Syntax hints:   -> wi 3   e. wcel 958   (_ wss 2047  <.cop 2411   class class class wbr 2619

This theorem is referenced by:  ssbri 2657

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-in 2051  df-ss 2053  df-br 2620

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