Theorem 3syld 57

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

Hypotheses

Ref	Expression
3syld.1	⊢ (𝜑 → (𝜓 → 𝜒))
3syld.2	⊢ (𝜑 → (𝜒 → 𝜃))
3syld.3	⊢ (𝜑 → (𝜃 → 𝜏))

Assertion

Ref	Expression
3syld	⊢ (𝜑 → (𝜓 → 𝜏))

Proof of Theorem 3syld

Step	Hyp	Ref	Expression
1		3syld.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2		3syld.2	. . 3 ⊢ (𝜑 → (𝜒 → 𝜃))
3	1, 2	syld 45	. 2 ⊢ (𝜑 → (𝜓 → 𝜃))
4		3syld.3	. 2 ⊢ (𝜑 → (𝜃 → 𝜏))
5	3, 4	syld 45	1 ⊢ (𝜑 → (𝜓 → 𝜏))

Syntax hints:   → wi 4

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

This theorem is referenced by:  xpfi  6811  fodjumkvlemres  7026  apreap  8342  msqge0  8371  cju  8712  facavg  10485  mulcn2  11074  coprm  11811  rpexp  11820  cnpnei  12377