Theorem sylanr2 634

Description: A syllogism inference. (Contributed by NM, 9-Apr-2005.)

Hypotheses

Ref	Expression
sylanr2.1	⊢ (φ → θ)
sylanr2.2	⊢ ((ψ ∧ (χ ∧ θ)) → τ)

Assertion

Ref	Expression
sylanr2	⊢ ((ψ ∧ (χ ∧ φ)) → τ)

Proof of Theorem sylanr2

Step	Hyp	Ref	Expression
1		sylanr2.1	. . 3 ⊢ (φ → θ)
2	1	anim2i 552	. 2 ⊢ ((χ ∧ φ) → (χ ∧ θ))
3		sylanr2.2	. 2 ⊢ ((ψ ∧ (χ ∧ θ)) → τ)
4	2, 3	sylan2 460	1 ⊢ ((ψ ∧ (χ ∧ φ)) → 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:  adantrrl  704  adantrrr  705