Theorem sylan9r 410

Description: Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
sylan9r.1	⊢ (𝜑 → (𝜓 → 𝜒))
sylan9r.2	⊢ (𝜃 → (𝜒 → 𝜏))

Assertion

Ref	Expression
sylan9r	⊢ ((𝜃 ∧ 𝜑) → (𝜓 → 𝜏))

Proof of Theorem sylan9r

Step	Hyp	Ref	Expression
1		sylan9r.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2		sylan9r.2	. . 3 ⊢ (𝜃 → (𝜒 → 𝜏))
3	1, 2	syl9r 73	. 2 ⊢ (𝜃 → (𝜑 → (𝜓 → 𝜏)))
4	3	imp 124	1 ⊢ ((𝜃 ∧ 𝜑) → (𝜓 → 𝜏))

Syntax hints:   → wi 4   ∧ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107

This theorem is referenced by:  spimt  1750  sbequi  1853  updjudhf  7146  genpcdl  7588  genpcuu  7589  iccsupr  10043  climuni  11460  tgcn  14454  metrest  14752