Theorem sylan9 389

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 66	. 2 ⊢ (𝜑 → (𝜃 → (𝜓 → 𝜏)))
4	3	imp 115	1 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → 𝜏))

Syntax hints:   → wi 4   ∧ wa 97

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100

This theorem is referenced by:  sbequi  1720  rspc2  2661  rspc3v  2665  trintssm  3870  copsexg  3981  chfnrn  5278  ffnfv  5323  f1elima  5412  smoel2  5918  th3q  6211  addnnnq0  6547  mulnnnq0  6548  addsrpr  6830  mulsrpr  6831  cau3lem  9710