Theorem sylan9 401

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 71	. 2 ⊢ (𝜑 → (𝜃 → (𝜓 → 𝜏)))
4	3	imp 122	1 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → 𝜏))

Syntax hints:   → wi 4   ∧ wa 102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105

This theorem is referenced by:  sbequi  1760  rspc2  2711  rspc3v  2716  trintssmOLD  3892  copsexg  3999  chfnrn  5299  ffnfv  5344  f1elima  5433  smoel2  5941  th3q  6234  addnnnq0  6639  mulnnnq0  6640  addsrpr  6922  mulsrpr  6923  cau3lem  10000