Theorem syld3an2 1229

Description: A syllogism inference. (Contributed by NM, 20-May-2007.)

Hypotheses

Ref	Expression
syld3an2.1	⊢ ((φ ∧ χ ∧ θ) → ψ)
syld3an2.2	⊢ ((φ ∧ ψ ∧ θ) → τ)

Assertion

Ref	Expression
syld3an2	⊢ ((φ ∧ χ ∧ θ) → τ)

Proof of Theorem syld3an2

Step	Hyp	Ref	Expression
1		syld3an2.1	. . . 4 ⊢ ((φ ∧ χ ∧ θ) → ψ)
2	1	3com23 1157	. . 3 ⊢ ((φ ∧ θ ∧ χ) → ψ)
3		syld3an2.2	. . . 4 ⊢ ((φ ∧ ψ ∧ θ) → τ)
4	3	3com23 1157	. . 3 ⊢ ((φ ∧ θ ∧ ψ) → τ)
5	2, 4	syld3an3 1227	. 2 ⊢ ((φ ∧ θ ∧ χ) → τ)
6	5	3com23 1157	1 ⊢ ((φ ∧ χ ∧ 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)