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  1887  rspc2  2922  rspc3v  2927  copsexg  4342  chfnrn  5767  ffnfv  5813  f1elima  5924  smoel2  6512  th3q  6852  fiintim  7166  addnnnq0  7712  mulnnnq0  7713  addsrpr  8008  mulsrpr  8009  cau3lem  11737  rescncf  15375