Theorem 3adant2l 1178

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 484	. 2 ⊢ ((𝜏 ∧ 𝜓) → 𝜓)
2		ad4ant3.1	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
3	1, 2	syl3an2 1164	1 ⊢ ((𝜑 ∧ (𝜏 ∧ 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:  axdc3lem4  10522  modexp  14287  lmmbr2  25312  ax5seglem1  28961  ax5seglem2  28962  nvaddsub4  30689  pl1cn  33901  athgt  39413  ltrncoelN  40100  ltrncoat  40101  trlcoabs  40678  tendoplcl2  40735  tendopltp  40737  dih1dimatlem0  41285  pellex  42791  fourierdlem42  46070