Theorem sstrd 1513

Description: Subclass transitivity deduction.

Hypotheses

Ref	Expression
sstrd.1	|- (ph -> A (_ B)
sstrd.2	|- (ph -> B (_ C)

Assertion

Ref	Expression
sstrd	|- (ph -> A (_ C)

Proof of Theorem sstrd

Step	Hyp	Ref	Expression
1		sstr 1511	. 2 |- ((A (_ B /\ B (_ C) -> A (_ C)
2		sstrd.1	. 2 |- (ph -> A (_ B)
3		sstrd.2	. 2 |- (ph -> B (_ C)
4	1, 2, 3	sylanc 361	1 |- (ph -> A (_ C)

Syntax hints:   -> wi 2   (_ wss 1487

This theorem is referenced by:  sylan9ss 1514  unss12 1630  sbthlem1 3349  shintcl 5294  shub1t 5353  ssmd2 5735  mdsymlem5 5780  sumdmdi 5785

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

metamath.org