Theorem sylan9ssr 2066

Description: A subclass transitivity deduction.

Hypotheses

Ref	Expression
sylan9ssr.1	|- (ph -> A (_ B)
sylan9ssr.2	|- (ps -> B (_ C)

Assertion

Ref	Expression
sylan9ssr	|- ((ps /\ ph) -> A (_ C)

Proof of Theorem sylan9ssr

Step	Hyp	Ref	Expression
1		sylan9ssr.1	. . 3 |- (ph -> A (_ B)
2		sylan9ssr.2	. . 3 |- (ps -> B (_ C)
3	1, 2	sylan9ss 2065	. 2 |- ((ph /\ ps) -> A (_ C)
4	3	ancoms 436	1 |- ((ps /\ ph) -> A (_ C)

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

This theorem is referenced by:  intssuni2 2546  cardinfima 4863  fgsb 10444

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

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