Theorem 3adant1l 1177

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 1163	1 ⊢ (((𝜏 ∧ 𝜑) ∧ 𝜓 ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086

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 1088

This theorem is referenced by:  ad5ant245  1363  cfsmolem  10223  axdc3lem4  10406  issubmnd  18688  mhmima  18752  rhmimasubrng  20475  maducoeval2  22527  cramerlem3  22576  restnlly  23369  efgh  26450  hasheuni  34075  matunitlindflem1  37610  pellex  42823  mendlmod  43178  disjf1o  45185  ssfiunibd  45307  mullimc  45614  mullimcf  45621  limclner  45649  limsupresxr  45764  liminfresxr  45765  sge0lefi  46396  isomenndlem  46528  hoicvr  46546  ovncvrrp  46562