Theorem 3adant1l 1178

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 484	. 2 ⊢ ((𝜏 ∧ 𝜑) → 𝜑)
2		ad4ant3.1	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
3	1, 2	syl3an1 1164	1 ⊢ (((𝜏 ∧ 𝜑) ∧ 𝜓 ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087

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  df-3an 1089

This theorem is referenced by:  ad5ant245  1364  cfsmolem  10192  axdc3lem4  10375  issubmnd  18698  mhmima  18762  rhmimasubrng  20511  maducoeval2  22596  cramerlem3  22645  restnlly  23438  efgh  26518  hasheuni  34262  matunitlindflem1  37861  pellex  43186  mendlmod  43540  disjf1o  45544  ssfiunibd  45665  mullimc  45970  mullimcf  45977  limclner  46003  limsupresxr  46118  liminfresxr  46119  sge0lefi  46750  isomenndlem  46882  hoicvr  46900  ovncvrrp  46916