Theorem syld3an1 1228

Description: A syllogism inference. (Contributed by NM, 7-Jul-2008.)

Hypotheses

Ref	Expression
syld3an1.1	⊢ ((χ ∧ ψ ∧ θ) → φ)
syld3an1.2	⊢ ((φ ∧ ψ ∧ θ) → τ)

Assertion

Ref	Expression
syld3an1	⊢ ((χ ∧ ψ ∧ θ) → τ)

Proof of Theorem syld3an1

Step	Hyp	Ref	Expression
1		syld3an1.1	. . . 4 ⊢ ((χ ∧ ψ ∧ θ) → φ)
2	1	3com13 1156	. . 3 ⊢ ((θ ∧ ψ ∧ χ) → φ)
3		syld3an1.2	. . . 4 ⊢ ((φ ∧ ψ ∧ θ) → τ)
4	3	3com13 1156	. . 3 ⊢ ((θ ∧ ψ ∧ φ) → τ)
5	2, 4	syld3an3 1227	. 2 ⊢ ((θ ∧ ψ ∧ χ) → τ)
6	5	3com13 1156	1 ⊢ ((χ ∧ ψ ∧ θ) → τ)

Syntax hints:   → wi 4   ∧ w3a 934

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  df-3an 936

This theorem is referenced by: (None)