Theorem 3adant2l 1163

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 1142	. . 3 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜃)
3	2	3adant1l 1161	. 2 ⊢ (((𝜏 ∧ 𝜓) ∧ 𝜑 ∧ 𝜒) → 𝜃)
4	3	3com12 1142	1 ⊢ ((𝜑 ∧ (𝜏 ∧ 𝜓) ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 102   ∧ w3a 919

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115  df-3an 921

This theorem is referenced by:  addassnqg  6572  mulassnqg  6574  prmuloc  6756  ltpopr  6785  addasssrg  6933  axaddass  7038