Theorem sylanr1 633

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

Hypotheses

Ref	Expression
sylanr1.1	⊢ (φ → χ)
sylanr1.2	⊢ ((ψ ∧ (χ ∧ θ)) → τ)

Assertion

Ref	Expression
sylanr1	⊢ ((ψ ∧ (φ ∧ θ)) → τ)

Proof of Theorem sylanr1

Step	Hyp	Ref	Expression
1		sylanr1.1	. . 3 ⊢ (φ → χ)
2	1	anim1i 551	. 2 ⊢ ((φ ∧ θ) → (χ ∧ θ))
3		sylanr1.2	. 2 ⊢ ((ψ ∧ (χ ∧ θ)) → τ)
4	2, 3	sylan2 460	1 ⊢ ((ψ ∧ (φ ∧ θ)) → τ)

Syntax hints:   → wi 4   ∧ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by:  adantrll  702  adantrlr  703