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  8282  pythagtrip  16812  nmolb2d  24613  nmoleub  24626  clwwisshclwwslem  29950  numclwwlk1lem2foa  30290  cvlcvr1  39339  4atlem12b  39612  dalawlem10  39881  dalawlem13  39884  dalawlem15  39886  osumcllem11N  39967  lhp2atne  40035  lhp2at0ne  40037  cdlemd  40208  ltrneq3  40209  cdleme7d  40247  cdlemeg49le  40512  cdleme  40561  cdlemg1a  40571  ltrniotavalbN  40585  cdlemg44  40734  cdlemk19  40870  cdlemk27-3  40908  cdlemk33N  40910  cdlemk34  40911  cdlemk49  40952  cdlemk53a  40956  cdlemk19u  40971  cdlemk56w  40974  dia2dimlem4  41068  dih1dimatlem0  41329  itsclc0yqe  48754  itsclinecirc0  48766  itsclinecirc0b  48767  inlinecirc02plem  48779