Theorem syl33anc 1385

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)
syl33anc.6	⊢ (𝜑 → 𝜁)
syl33anc.7	⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁)) → 𝜎)

Assertion

Ref	Expression
syl33anc	⊢ (𝜑 → 𝜎)

Proof of Theorem syl33anc

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		syl33anc.6	. 2 ⊢ (𝜑 → 𝜁)
8		syl33anc.7	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁)) → 𝜎)
9	4, 5, 6, 7, 8	syl13anc 1372	1 ⊢ (𝜑 → 𝜎)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087

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 1089

This theorem is referenced by:  xpord3inddlem  8195  initoeu2lem2  18082  mdetunilem9  22647  mdetuni0  22648  xmetrtri  24386  bl2in  24431  blhalf  24436  blssps  24455  blss  24456  blcld  24539  methaus  24554  metdstri  24892  metdscnlem  24896  metnrmlem3  24902  xlebnum  25016  pmltpclem1  25502  colinearalglem2  28940  axlowdim  28994  ssbnd  37748  totbndbnd  37749  heiborlem6  37776  2atm  39484  lplncvrlvol2  39572  dalem19  39639  paddasslem9  39785  pclclN  39848  pclfinN  39857  pclfinclN  39907  pexmidlem8N  39934  trlval3  40144  cdleme22b  40298  cdlemefr29bpre0N  40363  cdlemefr29clN  40364  cdlemefr32fvaN  40366  cdlemefr32fva1  40367  cdlemg31b0N  40651  cdlemg31b0a  40652  cdlemh  40774  dihmeetlem16N  41279  dihmeetlem18N  41281  dihmeetlem19N  41282  dihmeetlem20N  41283  hoidmvlelem1  46516