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 615	. 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜒 ∧ 𝜃))
3		sylanr1.2	. 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏)
4	2, 3	sylan2 593	1 ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜃)) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 395

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 207  df-an 396

This theorem is referenced by:  adantrll  722  adantrlr  723  sbthlem9  9003  unfi  9075  pczpre  16754  cpmadugsumlemF  22786  blsscls2  24414  rpvmasumlem  27420  leopmuli  32105  chirredlem1  32362  chirredlem3  32364  pibt2  37451  mhpind  42627  dvconstbi  44367  bccbc  44378  reccot  49790  rectan  49791  aacllem  49833