Theorem sylan9ss 2065

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 389	. 2 |- ((ph /\ ps) -> A (_ B)
3		sylan9ss.2	. . 3 |- (ps -> B (_ C)
4	3	adantl 388	. 2 |- ((ph /\ ps) -> B (_ C)
5	2, 4	sstrd 2064	1 |- ((ph /\ ps) -> A (_ C)

Syntax hints:   -> wi 3   /\ wa 223   (_ wss 2037

This theorem is referenced by:  sylan9ssr 2066  psstr 2140  unss12 2192  ss2in 2226  funssxp 3623  shslub 9273  chlej12 9313

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-in 2041  df-ss 2043

Copyright terms: Public domain