Theorem 3adant1l 1174

Description: Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 23-Jun-2022.)

Hypothesis

Ref	Expression
ad4ant3.1	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
3adant1l	⊢ (((𝜏 ∧ 𝜑) ∧ 𝜓 ∧ 𝜒) → 𝜃)

Proof of Theorem 3adant1l

Step	Hyp	Ref	Expression
1		simpr 483	. 2 ⊢ ((𝜏 ∧ 𝜑) → 𝜑)
2		ad4ant3.1	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
3	1, 2	syl3an1 1161	1 ⊢ (((𝜏 ∧ 𝜑) ∧ 𝜓 ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1085

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 206  df-an 395  df-3an 1087

This theorem is referenced by:  ad5ant245  1359  cfsmolem  10267  axdc3lem4  10450  issubmnd  18686  mhmima  18742  rhmimasubrng  20454  maducoeval2  22362  cramerlem3  22411  restnlly  23206  efgh  26286  hasheuni  33381  matunitlindflem1  36787  pellex  41875  mendlmod  42237  disjf1o  44188  ssfiunibd  44317  mullimc  44630  mullimcf  44637  limclner  44665  limsupresxr  44780  liminfresxr  44781  sge0lefi  45412  isomenndlem  45544  hoicvr  45562  ovncvrrp  45578