Theorem syl9 66

Description: A nested syllogism inference with different antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)

Hypotheses

Ref	Expression
syl9.1	⊢ (φ → (ψ → χ))
syl9.2	⊢ (θ → (χ → τ))

Assertion

Ref	Expression
syl9	⊢ (φ → (θ → (ψ → τ)))

Proof of Theorem syl9

Step	Hyp	Ref	Expression
1		syl9.1	. 2 ⊢ (φ → (ψ → χ))
2		syl9.2	. . 3 ⊢ (θ → (χ → τ))
3	2	a1i 10	. 2 ⊢ (φ → (θ → (χ → τ)))
4	1, 3	syl5d 62	1 ⊢ (φ → (θ → (ψ → τ)))

Syntax hints:   → wi 4

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

This theorem is referenced by:  syl9r  67  com23  72  sylan9  638  19.21t  1795  19.21tOLD  1863  sbequi  2059  sbal1  2126  reuss2  3536  reupick  3540  ssfin  4471  iss  5001