Theorem 3adant1l 1208

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
3adant1l	⊢ (((𝜏 ∧ 𝜑) ∧ 𝜓 ∧ 𝜒) → 𝜃)

Proof of Theorem 3adant1l

Step	Hyp	Ref	Expression
1		3adant1l.1	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
2	1	3expb 1182	. . 3 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)
3	2	adantll 467	. 2 ⊢ (((𝜏 ∧ 𝜑) ∧ (𝜓 ∧ 𝜒)) → 𝜃)
4	3	3impb 1177	1 ⊢ (((𝜏 ∧ 𝜑) ∧ 𝜓 ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 964

This theorem is referenced by:  3adant2l  1210  3adant3l  1212  tfrcl  6254  addassnqg  7183  mulassnqg  7185  addasssrg  7557  axaddass  7673