Theorem 3syld 57

Description: Triple syllogism deduction. (Contributed by Jeff Hankins, 4-Aug-2009.)

Hypotheses

Ref	Expression
3syld.1	|- ( ph -> ( ps -> ch ) )
3syld.2	|- ( ph -> ( ch -> th ) )
3syld.3	|- ( ph -> ( th -> ta ) )

Assertion

Ref	Expression
3syld	|- ( ph -> ( ps -> ta ) )

Proof of Theorem 3syld

Step	Hyp	Ref	Expression
1		3syld.1	. . 3 |- ( ph -> ( ps -> ch ) )
2		3syld.2	. . 3 |- ( ph -> ( ch -> th ) )
3	1, 2	syld 45	. 2 |- ( ph -> ( ps -> th ) )
4		3syld.3	. 2 |- ( ph -> ( th -> ta ) )
5	3, 4	syld 45	1 |- ( ph -> ( ps -> ta ) )

This proof depends on syntax axioms:   -> wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is used by:  xpfi  7239  fodjumkvlemres  7499  enmkvlem  7501  apreap  8915  msqge0  8944  cju  9291  facavg  11184  mulcn2  12078  coprm  12922  rpexp  12931  cnpnei  15320  lgseisenlem2  16190  uspgr2wlkeq  16606  ismkvnnlem  17102