Theorem syl311anc 1386

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

Hypotheses

Ref	Expression
syl3anc.1	⊢ (𝜑 → 𝜓)
syl3anc.2	⊢ (𝜑 → 𝜒)
syl3anc.3	⊢ (𝜑 → 𝜃)
syl3Xanc.4	⊢ (𝜑 → 𝜏)
syl23anc.5	⊢ (𝜑 → 𝜂)
syl311anc.6	⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏 ∧ 𝜂) → 𝜁)

Assertion

Ref	Expression
syl311anc	⊢ (𝜑 → 𝜁)

Proof of Theorem syl311anc

Step	Hyp	Ref	Expression
1		syl3anc.1	. . 3 ⊢ (𝜑 → 𝜓)
2		syl3anc.2	. . 3 ⊢ (𝜑 → 𝜒)
3		syl3anc.3	. . 3 ⊢ (𝜑 → 𝜃)
4	1, 2, 3	3jca 1128	. 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))
5		syl3Xanc.4	. 2 ⊢ (𝜑 → 𝜏)
6		syl23anc.5	. 2 ⊢ (𝜑 → 𝜂)
7		syl311anc.6	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏 ∧ 𝜂) → 𝜁)
8	4, 5, 6, 7	syl3anc 1373	1 ⊢ (𝜑 → 𝜁)

Syntax hints:   → wi 4   ∧ 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:  syl312anc  1393  syl321anc  1394  syl313anc  1396  syl331anc  1397  fprlem1  8225  pythagtrip  16741  nmolb2d  24628  nmoleub  24641  clwwisshclwwslem  29986  numclwwlk1lem2foa  30326  cvlcvr1  39378  4atlem12b  39650  dalawlem10  39919  dalawlem13  39922  dalawlem15  39924  osumcllem11N  40005  lhp2atne  40073  lhp2at0ne  40075  cdlemd  40246  ltrneq3  40247  cdleme7d  40285  cdlemeg49le  40550  cdleme  40599  cdlemg1a  40609  ltrniotavalbN  40623  cdlemg44  40772  cdlemk19  40908  cdlemk27-3  40946  cdlemk33N  40948  cdlemk34  40949  cdlemk49  40990  cdlemk53a  40994  cdlemk19u  41009  cdlemk56w  41012  dia2dimlem4  41106  dih1dimatlem0  41367  itsclc0yqe  48793  itsclinecirc0  48805  itsclinecirc0b  48806  inlinecirc02plem  48818