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  4245  funimaexglem  5464  smoiun  6572  en2  7112  findcard2  7193  ctssdc  7454  mkvprop  7499  ltexprlemrl  7978  archsr  8150  elfz0ubfz0  10543  ctinf  13372  wlkres  16742