Theorem mpsylsyld 70

Description: Modus ponens combined with a double syllogism inference. (Contributed by Alan Sare, 22-Jul-2012.)

Hypotheses

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

Assertion

Ref	Expression
mpsylsyld	⊢ (𝜓 → (𝜒 → 𝜏))

Proof of Theorem mpsylsyld

Step	Hyp	Ref	Expression
1		mpsylsyld.1	. . 3 ⊢ 𝜑
2	1	a1i 11	. 2 ⊢ (𝜓 → 𝜑)
3		mpsylsyld.2	. 2 ⊢ (𝜓 → (𝜒 → 𝜃))
4		mpsylsyld.3	. 2 ⊢ (𝜑 → (𝜃 → 𝜏))
5	2, 3, 4	sylsyld 62	1 ⊢ (𝜓 → (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:  r1sdom  9744  r1ord3g  9749  r1ord2  9751  rlimclim  15604  vk15.4j  45265  onfrALTlem3  45281  ee02an  45436  usgrexmpl12ngric  48831  usgrexmpl12ngrlic  48832