Theorem sylan9r 639

Description: Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
sylan9r.1	⊢ (φ → (ψ → χ))
sylan9r.2	⊢ (θ → (χ → τ))

Assertion

Ref	Expression
sylan9r	⊢ ((θ ∧ φ) → (ψ → τ))

Proof of Theorem sylan9r

Step	Hyp	Ref	Expression
1		sylan9r.1	. . 3 ⊢ (φ → (ψ → χ))
2		sylan9r.2	. . 3 ⊢ (θ → (χ → τ))
3	1, 2	syl9r 67	. 2 ⊢ (θ → (φ → (ψ → τ)))
4	3	imp 418	1 ⊢ ((θ ∧ φ) → (ψ → τ))

Syntax hints:   → wi 4   ∧ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by:  sbequi  2059  opkth1g  4131  ncfinraise  4482  funun  5147