Theorem 3adant1l 1195

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 490	. 2 ⊢ ((𝜏 ∧ 𝜑) → 𝜑)
2		ad4ant3.1	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
3	1, 2	syl3an1 1181	1 ⊢ (((𝜏 ∧ 𝜑) ∧ 𝜓 ∧ 𝜒) → 𝜃)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103

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  df-3an 1105

This theorem is used by:  ad5ant245  1384  cfsmolem  10329  axdc3lem4  10512  issubmnd  18933  mhmima  19001  rhmimasubrng  20798  maducoeval2  22935  matunitlindflem1  22974  cramerlem3  22987  restnlly  23781  efgh  26851  hasheuni  34699  pellex  43795  mendlmod  44149  disjf1o  46149  ssfiunibd  46268  mullimc  46572  mullimcf  46579  limclner  46605  limsupresxr  46720  liminfresxr  46721  sge0lefi  47352  isomenndlem  47484  hoicvr  47502  ovncvrrp  47518