Theorem syl312anc 1393

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 511	. 2 ⊢ (𝜑 → (𝜂 ∧ 𝜁))
8		syl312anc.7	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏 ∧ (𝜂 ∧ 𝜁)) → 𝜎)
9	1, 2, 3, 4, 7, 8	syl311anc 1386	1 ⊢ (𝜑 → 𝜎)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086

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 1088

This theorem is referenced by:  pythagtriplem19  16804  cdleme27cl  40360  cdlemefs27cl  40407  cdleme32fvcl  40434  cdlemg16ALTN  40652  cdlemg27a  40686  cdlemg31c  40693  cdlemg39  40710  cdlemk11ta  40923  cdlemk19ylem  40924  cdlemk11tc  40939  cdlemk45  40941  dihmeetlem12N  41312  dihjatc  41411  flt4lem5c  42642  flt4lem5d  42643  flt4lem5e  42644