Theorem sylan9 407

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 123	1 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → 𝜏))

Syntax hints:   → wi 4   ∧ wa 103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106

This theorem is referenced by:  sbequi  1812  rspc2  2804  rspc3v  2809  copsexg  4174  chfnrn  5539  ffnfv  5586  f1elima  5682  smoel2  6208  th3q  6542  fiintim  6825  addnnnq0  7281  mulnnnq0  7282  addsrpr  7577  mulsrpr  7578  cau3lem  10918  rescncf  12776