Theorem sylan9ss 2628

Description: A subclass transitivity deduction. (The proof was shortened by Andrew Salmon, 14-Jun-2011.)

Hypotheses

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

Assertion

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

Proof of Theorem sylan9ss

Step	Hyp	Ref	Expression
1		sstr 2625	. 2 |- ((A C_ B /\ B C_ C) -> A C_ C)
2		sylan9ss.1	. 2 |- (ph -> A C_ B)
3		sylan9ss.2	. 2 |- (ps -> B C_ C)
4	1, 2, 3	syl2an 503	1 |- ((ph /\ ps) -> A C_ C)

Syntax hints:   -> wi 3   /\ wa 240   C_ wss 2593

This theorem is referenced by:  sylan9ssr 2630  psstr 2714  unss12 2778  ss2in 2820  relrelss 4417  funssxp 4577  shslubi 10991  chlej12i 11031  bnj1011 12876

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-in 2603  df-ss 2605

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