Theorem sylanl1 631

Description: A syllogism inference. (Contributed by NM, 10-Mar-2005.)

Hypotheses

Ref	Expression
sylanl1.1	⊢ (φ → ψ)
sylanl1.2	⊢ (((ψ ∧ χ) ∧ θ) → τ)

Assertion

Ref	Expression
sylanl1	⊢ (((φ ∧ χ) ∧ θ) → τ)

Proof of Theorem sylanl1

Step	Hyp	Ref	Expression
1		sylanl1.1	. . 3 ⊢ (φ → ψ)
2	1	anim1i 551	. 2 ⊢ ((φ ∧ χ) → (ψ ∧ χ))
3		sylanl1.2	. 2 ⊢ (((ψ ∧ χ) ∧ θ) → τ)
4	2, 3	sylan 457	1 ⊢ (((φ ∧ χ) ∧ 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:  adantlll  698  adantllr  699  isocnv  5492