Theorem syl33anc 1412

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 1146	. 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 1399	1 ⊢ (𝜑 → 𝜎)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105

This theorem is used by:  xpord3inddlem  8173  initoeu2lem2  18195  mdetunilem9  22940  mdetuni0  22941  xmetrtri  24679  bl2in  24724  blhalf  24729  blssps  24748  blss  24749  blcld  24829  methaus  24844  metdstri  25176  metdscnlem  25180  metnrmlem3  25186  xlebnum  25291  pmltpclem1  25774  bdayfinbndlem1  28858  colinearalglem2  29490  axlowdim  29544  ssbnd  38815  totbndbnd  38816  heiborlem6  38843  2atm  40677  lplncvrlvol2  40765  dalem19  40832  paddasslem9  40978  pclclN  41041  pclfinN  41050  pclfinclN  41100  pexmidlem8N  41127  trlval3  41337  cdleme22b  41491  cdlemefr29bpre0N  41556  cdlemefr29clN  41557  cdlemefr32fvaN  41559  cdlemefr32fva1  41560  cdlemg31b0N  41844  cdlemg31b0a  41845  cdlemh  41967  dihmeetlem16N  42472  dihmeetlem18N  42474  dihmeetlem19N  42475  dihmeetlem20N  42476  hoidmvlelem1  47698  veroquadnolindfd  51153