Theorem syl9r 67

Description: A nested syllogism inference with different antecedents. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
syl9r.1	⊢ (φ → (ψ → χ))
syl9r.2	⊢ (θ → (χ → τ))

Assertion

Ref	Expression
syl9r	⊢ (θ → (φ → (ψ → τ)))

Proof of Theorem syl9r

Step	Hyp	Ref	Expression
1		syl9r.1	. . 3 ⊢ (φ → (ψ → χ))
2		syl9r.2	. . 3 ⊢ (θ → (χ → τ))
3	1, 2	syl9 66	. 2 ⊢ (φ → (θ → (ψ → τ)))
4	3	com12 27	1 ⊢ (θ → (φ → (ψ → τ)))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is used by:  sylan9r  639  19.23t  1800  nfimd  1808  spfinsfincl  4540  fununi  5161  funimass3  5405  weds  5939