Theorem sylanr1 695

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 627	. 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜒 ∧ 𝜃))
3		sylanr1.2	. 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏)
4	2, 3	sylan2 605	1 ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜃)) → 𝜏)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  adantrll  735  adantrlr  736  sbthlem9  9152  unfi  9224  pczpre  17069  cpmadugsumlemF  23249  blsscls2  24878  rpvmasumlem  27871  leopmuli  32792  chirredlem1  33049  chirredlem3  33051  pibt2  38608  mhpind  43888  dvconstbi  45579  bccbc  45590  reccot  51279  rectan  51280  aacllem  51367