Theorem syl312anc 1409

Description: Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)

Hypotheses

Ref	Expression
syl3anc.1	⊢ (𝜑 → 𝜓)
syl3anc.2	⊢ (𝜑 → 𝜒)
syl3anc.3	⊢ (𝜑 → 𝜃)
syl3Xanc.4	⊢ (𝜑 → 𝜏)
syl23anc.5	⊢ (𝜑 → 𝜂)
syl33anc.6	⊢ (𝜑 → 𝜁)
syl312anc.7	⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏 ∧ (𝜂 ∧ 𝜁)) → 𝜎)

Assertion

Ref	Expression
syl312anc	⊢ (𝜑 → 𝜎)

Proof of Theorem syl312anc

Step	Hyp	Ref	Expression
1		syl3anc.1	. 2 ⊢ (𝜑 → 𝜓)
2		syl3anc.2	. 2 ⊢ (𝜑 → 𝜒)
3		syl3anc.3	. 2 ⊢ (𝜑 → 𝜃)
4		syl3Xanc.4	. 2 ⊢ (𝜑 → 𝜏)
5		syl23anc.5	. . 3 ⊢ (𝜑 → 𝜂)
6		syl33anc.6	. . 3 ⊢ (𝜑 → 𝜁)
7	5, 6	jca 519	. 2 ⊢ (𝜑 → (𝜂 ∧ 𝜁))
8		syl312anc.7	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏 ∧ (𝜂 ∧ 𝜁)) → 𝜎)
9	1, 2, 3, 4, 7, 8	syl311anc 1402	1 ⊢ (𝜑 → 𝜎)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097

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 209  df-an 400  df-3an 1099

This theorem is referenced by:  pythagtriplem19  16852  cdleme27cl  40954  cdlemefs27cl  41001  cdleme32fvcl  41028  cdlemg16ALTN  41246  cdlemg27a  41280  cdlemg31c  41287  cdlemg39  41304  cdlemk11ta  41517  cdlemk19ylem  41518  cdlemk11tc  41533  cdlemk45  41535  dihmeetlem12N  41906  dihjatc  42005  flt4lem5c  43200  flt4lem5d  43201  flt4lem5e  43202