Theorem 3adant2l 1197

Description: Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 25-Jun-2022.)

Hypothesis

Ref	Expression
ad4ant3.1	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
3adant2l	⊢ ((𝜑 ∧ (𝜏 ∧ 𝜓) ∧ 𝜒) → 𝜃)

Proof of Theorem 3adant2l

Step	Hyp	Ref	Expression
1		simpr 490	. 2 ⊢ ((𝜏 ∧ 𝜓) → 𝜓)
2		ad4ant3.1	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
3	1, 2	syl3an2 1182	1 ⊢ ((𝜑 ∧ (𝜏 ∧ 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:  axdc3lem4  10531  modexp  14382  lmmbr2  25580  ax5seglem1  29506  ax5seglem2  29507  nvaddsub4  31259  pl1cn  34629  eldisjs6  39915  athgt  40556  ltrncoelN  41243  ltrncoat  41244  trlcoabs  41821  tendoplcl2  41878  tendopltp  41880  dih1dimatlem0  42428  pellex  43884  fourierdlem42  47201