Theorem sylanl2 632

Description: A syllogism inference. (Contributed by NM, 1-Jan-2005.)

Hypotheses

Ref	Expression
sylanl2.1	⊢ (φ → χ)
sylanl2.2	⊢ (((ψ ∧ χ) ∧ θ) → τ)

Assertion

Ref	Expression
sylanl2	⊢ (((ψ ∧ φ) ∧ θ) → τ)

Proof of Theorem sylanl2

Step	Hyp	Ref	Expression
1		sylanl2.1	. . 3 ⊢ (φ → χ)
2	1	anim2i 552	. 2 ⊢ ((ψ ∧ φ) → (ψ ∧ χ))
3		sylanl2.2	. 2 ⊢ (((ψ ∧ χ) ∧ θ) → τ)
4	2, 3	sylan 457	1 ⊢ (((ψ ∧ φ) ∧ 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:  mpanlr1  667  adantlrl  700  adantlrr  701  ncssfin  6152  spacind  6288