Theorem 3adant1l 1176

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 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  1361  cfsmolem  10339  axdc3lem4  10522  issubmnd  18799  mhmima  18860  rhmimasubrng  20592  maducoeval2  22667  cramerlem3  22716  restnlly  23511  efgh  26601  hasheuni  34049  matunitlindflem1  37576  pellex  42791  mendlmod  43150  disjf1o  45098  ssfiunibd  45224  mullimc  45537  mullimcf  45544  limclner  45572  limsupresxr  45687  liminfresxr  45688  sge0lefi  46319  isomenndlem  46451  hoicvr  46469  ovncvrrp  46485