Theorem 3adant2l 1312

Description: Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)

Hypothesis

Ref	Expression
3adant1l.1	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Assertion

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

Proof of Theorem 3adant2l

Step	Hyp	Ref	Expression
1		3adant1l.1	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
2	1	3com12 1261	. . 3 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜃)
3	2	3adant1l 1310	. 2 ⊢ (((𝜏 ∧ 𝜓) ∧ 𝜑 ∧ 𝜒) → 𝜃)
4	3	3com12 1261	1 ⊢ ((𝜑 ∧ (𝜏 ∧ 𝜓) ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031

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 196  df-an 385  df-3an 1033

This theorem is referenced by:  axdc3lem4  9158  modexp  12861  lmmbr2  22865  ax5seglem1  25608  ax5seglem2  25609  nvaddsub4  26896  pl1cn  29329  athgt  33760  ltrncoelN  34447  ltrncoat  34448  trlcoabs  35027  tendoplcl2  35084  tendopltp  35086  dih1dimatlem0  35635  pellex  36417  fourierdlem42  39042