Theorem sylan9ssr 2079

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 2078	. 2 |- ((ph /\ ps) -> A (_ C)
4	3	ancoms 438	1 |- ((ps /\ ph) -> A (_ C)

Syntax hints:   -> wi 3   /\ wa 223   (_ wss 2050

This theorem is referenced by:  intssuni2 2560  cardinfima 4902  fgsb 10555

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-in 2054  df-ss 2056

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