Theorem sylanr1 682

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 590	. 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜒 ∧ 𝜃))
3		sylanr1.2	. 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏)
4	2, 3	sylan2 490	1 ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜃)) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 383

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 196  df-an 385

This theorem is referenced by:  adantrll  754  adantrlr  755  sbthlem9  7963  pczpre  15390  cpmadugsumlemF  20500  blsscls2  22119  rpvmasumlem  24976  leopmuli  28376  chirredlem1  28633  chirredlem3  28635  dvconstbi  37555  bccbc  37566  reccot  42298  rectan  42299  aacllem  42356