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  10158  axdc3lem4  10341  issubmnd  18666  mhmima  18730  rhmimasubrng  20479  maducoeval2  22553  cramerlem3  22602  restnlly  23395  efgh  26475  hasheuni  34093  matunitlindflem1  37655  pellex  42867  mendlmod  43221  disjf1o  45227  ssfiunibd  45349  mullimc  45655  mullimcf  45662  limclner  45688  limsupresxr  45803  liminfresxr  45804  sge0lefi  46435  isomenndlem  46567  hoicvr  46585  ovncvrrp  46601