Theorem syl33anc 1383

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 1126	. 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 1370	1 ⊢ (𝜑 → 𝜎)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085

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 206  df-an 396  df-3an 1087

This theorem is referenced by:  initoeu2lem2  17646  mdetunilem9  21677  mdetuni0  21678  xmetrtri  23416  bl2in  23461  blhalf  23466  blssps  23485  blss  23486  blcld  23567  methaus  23582  metdstri  23920  metdscnlem  23924  metnrmlem3  23930  xlebnum  24034  pmltpclem1  24517  colinearalglem2  27178  axlowdim  27232  ssbnd  35873  totbndbnd  35874  heiborlem6  35901  2atm  37468  lplncvrlvol2  37556  dalem19  37623  paddasslem9  37769  pclclN  37832  pclfinN  37841  pclfinclN  37891  pexmidlem8N  37918  trlval3  38128  cdleme22b  38282  cdlemefr29bpre0N  38347  cdlemefr29clN  38348  cdlemefr32fvaN  38350  cdlemefr32fva1  38351  cdlemg31b0N  38635  cdlemg31b0a  38636  cdlemh  38758  dihmeetlem16N  39263  dihmeetlem18N  39265  dihmeetlem19N  39266  dihmeetlem20N  39267  hoidmvlelem1  44023