Theorem syl3an2 1216

Description: A syllogism inference. (Contributed by NM, 22-Aug-1995.)

Hypotheses

Ref	Expression
syl3an2.1	⊢ (φ → χ)
syl3an2.2	⊢ ((ψ ∧ χ ∧ θ) → τ)

Assertion

Ref	Expression
syl3an2	⊢ ((ψ ∧ φ ∧ θ) → τ)

Proof of Theorem syl3an2

Step	Hyp	Ref	Expression
1		syl3an2.1	. . 3 ⊢ (φ → χ)
2		syl3an2.2	. . . 4 ⊢ ((ψ ∧ χ ∧ θ) → τ)
3	2	3exp 1150	. . 3 ⊢ (ψ → (χ → (θ → τ)))
4	1, 3	syl5 28	. 2 ⊢ (ψ → (φ → (θ → τ)))
5	4	3imp 1145	1 ⊢ ((ψ ∧ φ ∧ 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:  syl3an2b  1219  syl3an2br  1222  syl3anl2  1231