Theorem syl2and 469

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

Hypotheses

Ref	Expression
syl2and.1	⊢ (φ → (ψ → χ))
syl2and.2	⊢ (φ → (θ → τ))
syl2and.3	⊢ (φ → ((χ ∧ τ) → η))

Assertion

Ref	Expression
syl2and	⊢ (φ → ((ψ ∧ θ) → η))

Proof of Theorem syl2and

Step	Hyp	Ref	Expression
1		syl2and.1	. 2 ⊢ (φ → (ψ → χ))
2		syl2and.2	. . 3 ⊢ (φ → (θ → τ))
3		syl2and.3	. . 3 ⊢ (φ → ((χ ∧ τ) → η))
4	2, 3	sylan2d 468	. 2 ⊢ (φ → ((χ ∧ θ) → η))
5	1, 4	syland 467	1 ⊢ (φ → ((ψ ∧ θ) → η))

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:  anim12d  546  tfin11  4494