Theorem sylani 635

Description: A syllogism inference. (Contributed by NM, 2-May-1996.)

Hypotheses

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

Assertion

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

Proof of Theorem sylani

Step	Hyp	Ref	Expression
1		sylani.1	. . 3 ⊢ (φ → χ)
2	1	a1i 10	. 2 ⊢ (ψ → (φ → χ))
3		sylani.2	. 2 ⊢ (ψ → ((χ ∧ θ) → τ))
4	2, 3	syland 467	1 ⊢ (ψ → ((φ ∧ θ) → τ))

Syntax hints:   → wi 4   ∧ wa 358

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

This theorem is referenced by:  syl2ani  637