Theorem syl2ani 637

Description: A syllogism inference. (Contributed by NM, 3-Aug-1999.)

Hypotheses

Ref	Expression
syl2ani.1	⊢ (φ → χ)
syl2ani.2	⊢ (η → θ)
syl2ani.3	⊢ (ψ → ((χ ∧ θ) → τ))

Assertion

Ref	Expression
syl2ani	⊢ (ψ → ((φ ∧ η) → τ))

Proof of Theorem syl2ani

Step	Hyp	Ref	Expression
1		syl2ani.1	. 2 ⊢ (φ → χ)
2		syl2ani.2	. . 3 ⊢ (η → θ)
3		syl2ani.3	. . 3 ⊢ (ψ → ((χ ∧ θ) → τ))
4	2, 3	sylan2i 636	. 2 ⊢ (ψ → ((χ ∧ η) → τ))
5	1, 4	sylani 635	1 ⊢ (ψ → ((φ ∧ η) → τ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360

This theorem is used by: (None)