Theorem sylan9 409

Description: Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Hypotheses

Ref	Expression
sylan9.1	⊢ (𝜑 → (𝜓 → 𝜒))
sylan9.2	⊢ (𝜃 → (𝜒 → 𝜏))

Assertion

Ref	Expression
sylan9	⊢ ((𝜑 ∧ 𝜃) → (𝜓 → 𝜏))

Proof of Theorem sylan9

Step	Hyp	Ref	Expression
1		sylan9.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2		sylan9.2	. . 3 ⊢ (𝜃 → (𝜒 → 𝜏))
3	1, 2	syl9 72	. 2 ⊢ (𝜑 → (𝜃 → (𝜓 → 𝜏)))
4	3	imp 124	1 ⊢ ((𝜑 ∧ 𝜃) → (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:  sbequi  1850  rspc2  2875  rspc3v  2880  copsexg  4273  chfnrn  5669  ffnfv  5716  f1elima  5816  smoel2  6356  th3q  6694  fiintim  6985  addnnnq0  7509  mulnnnq0  7510  addsrpr  7805  mulsrpr  7806  cau3lem  11258  rescncf  14736