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))

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:  ax10o  1767  a16g  1917  rspc2vd  3216  trintssm  4247  funimaexglem  5467  smoiun  6577  en2  7118  findcard2  7199  ctssdc  7460  mkvprop  7505  ltexprlemrl  7984  archsr  8156  elfz0ubfz0  10549  ctinf  13379  mvrf1fi  15231  wlkres  16858