Theorem sylsyld 58

Description: A double syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.)

Hypotheses

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

Assertion

Ref	Expression
sylsyld	⊢ (𝜑 → (𝜒 → 𝜏))

Proof of Theorem sylsyld

Step	Hyp	Ref	Expression
1		sylsyld.2	. 2 ⊢ (𝜑 → (𝜒 → 𝜃))
2		sylsyld.1	. . 3 ⊢ (𝜑 → 𝜓)
3		sylsyld.3	. . 3 ⊢ (𝜓 → (𝜃 → 𝜏))
4	2, 3	syl 14	. 2 ⊢ (𝜑 → (𝜃 → 𝜏))
5	1, 4	syld 45	1 ⊢ (𝜑 → (𝜒 → 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:  ax10o  1708  a16g  1857  rspc2vd  3117  trintssm  4101  funimaexglem  5279  smoiun  6278  findcard2  6864  ctssdc  7087  mkvprop  7131  ltexprlemrl  7561  archsr  7733  elfz0ubfz0  10070  ctinf  12374