Theorem 3adant2l 1185

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 485	. 2 ⊢ ((𝜏 ∧ 𝜓) → 𝜓)
2		ad4ant3.1	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
3	1, 2	syl3an2 1170	1 ⊢ ((𝜑 ∧ (𝜏 ∧ 𝜓) ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092

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 208  df-an 397  df-3an 1094

This theorem is referenced by:  axdc3lem4  10373  modexp  14198  lmmbr2  25251  ax5seglem1  29022  ax5seglem2  29023  nvaddsub4  30753  pl1cn  34146  eldisjs6  39314  athgt  39955  ltrncoelN  40642  ltrncoat  40643  trlcoabs  41220  tendoplcl2  41277  tendopltp  41279  dih1dimatlem0  41827  pellex  43287  fourierdlem42  46599