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  8242  pythagtrip  16762  nmolb2d  24662  nmoleub  24675  clwwisshclwwslem  30089  numclwwlk1lem2foa  30429  cvlcvr1  39599  4atlem12b  39871  dalawlem10  40140  dalawlem13  40143  dalawlem15  40145  osumcllem11N  40226  lhp2atne  40294  lhp2at0ne  40296  cdlemd  40467  ltrneq3  40468  cdleme7d  40506  cdlemeg49le  40771  cdleme  40820  cdlemg1a  40830  ltrniotavalbN  40844  cdlemg44  40993  cdlemk19  41129  cdlemk27-3  41167  cdlemk33N  41169  cdlemk34  41170  cdlemk49  41211  cdlemk53a  41215  cdlemk19u  41230  cdlemk56w  41233  dia2dimlem4  41327  dih1dimatlem0  41588  itsclc0yqe  49007  itsclinecirc0  49019  itsclinecirc0b  49020  inlinecirc02plem  49032