Theorem sylanr1 672

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 608	. 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜒 ∧ 𝜃))
3		sylanr1.2	. 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏)
4	2, 3	sylan2 586	1 ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜃)) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 386

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 199  df-an 387

This theorem is referenced by:  adantrll  713  adantrlr  714  sbthlem9  8353  pczpre  15930  cpmadugsumlemF  21058  blsscls2  22686  rpvmasumlem  25596  leopmuli  29543  chirredlem1  29800  chirredlem3  29802  dvconstbi  39372  bccbc  39383  reccot  43411  rectan  43412  aacllem  43457