Theorem sylan9ss 2071

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 2070	1 |- ((ph /\ ps) -> A (_ C)

Syntax hints:   -> wi 3   /\ wa 223   (_ wss 2043

This theorem is referenced by:  sylan9ssr 2072  psstr 2146  unss12 2198  ss2in 2232  funssxp 3629  shslub 9296  chlej12 9336

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-in 2047  df-ss 2049

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