Theorem sylan9ss 1514

Description: A subclass transitivity deduction.

Hypotheses

Ref	Expression
sylan9ss.1	|- (ph -> A (_ B)
sylan9ss.2	|- (ps -> B (_ C)

Assertion

Ref	Expression
sylan9ss	|- ((ph /\ ps) -> A (_ C)

Proof of Theorem sylan9ss

Step	Hyp	Ref	Expression
1		sylan9ss.1	. . 3 |- (ph -> A (_ B)
2	1	adantr 306	. 2 |- ((ph /\ ps) -> A (_ B)
3		sylan9ss.2	. . 3 |- (ps -> B (_ C)
4	3	adantl 305	. 2 |- ((ph /\ ps) -> B (_ C)
5	2, 4	sstrd 1513	1 |- ((ph /\ ps) -> A (_ C)

Syntax hints:   -> wi 2   /\ wa 196   (_ wss 1487

This theorem is referenced by:  sylan9ssr 1515  psstr 1574  ss2in 1663  shslub 5359  chlej12 5396

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

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