Theorem syland 467

Description: A syllogism deduction. (Contributed by NM, 15-Dec-2004.)

Hypotheses

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

Assertion

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

Proof of Theorem syland

Step	Hyp	Ref	Expression
1		syland.1	. . 3 ⊢ (φ → (ψ → χ))
2		syland.2	. . . 4 ⊢ (φ → ((χ ∧ θ) → τ))
3	2	exp3a 425	. . 3 ⊢ (φ → (χ → (θ → τ)))
4	1, 3	syld 40	. 2 ⊢ (φ → (ψ → (θ → τ)))
5	4	imp3a 420	1 ⊢ (φ → ((ψ ∧ θ) → 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:  sylan2d  468  syl2and  469  sylani  635