Theorem syldd 72

Description: Nested syllogism deduction. Deduction associated with syld 47. Double deduction associated with syl 17. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 11-May-2013.)

Hypotheses

Ref	Expression
syldd.1	|- ( ph -> ( ps -> ( ch -> th ) ) )
syldd.2	|- ( ph -> ( ps -> ( th -> ta ) ) )

Assertion

Ref	Expression
syldd	|- ( ph -> ( ps -> ( ch -> ta ) ) )

Proof of Theorem syldd

Step	Hyp	Ref	Expression
1		syldd.2	. 2 |- ( ph -> ( ps -> ( th -> ta ) ) )
2		syldd.1	. 2 |- ( ph -> ( ps -> ( ch -> th ) ) )
3		imim2 58	. 2 |- ( ( th -> ta ) -> ( ( ch -> th ) -> ( ch -> ta ) ) )
4	1, 2, 3	syl6c 70	1 |- ( ph -> ( ps -> ( ch -> ta ) ) )

Syntax hints:   -> wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is referenced by:  syl5d  73  syl6d  75  syl10  79  tfinds  7059  tz7.49  7540  dffi2  8329  ordiso2  8420  rankuni2b  8716  oddprmdvds  15607  brbtwn2  25785  soseq  31751  bj-exalims  32613  prtlem60  34137  lvoli2  34867