Theorem syl33anc 1388

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 1129	. 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 1375	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  8097  initoeu2lem2  17973  mdetunilem9  22595  mdetuni0  22596  xmetrtri  24330  bl2in  24375  blhalf  24380  blssps  24399  blss  24400  blcld  24480  methaus  24495  metdstri  24827  metdscnlem  24831  metnrmlem3  24837  xlebnum  24942  pmltpclem1  25425  bdayfinbndlem1  28473  colinearalglem2  28990  axlowdim  29044  ssbnd  38123  totbndbnd  38124  heiborlem6  38151  2atm  39987  lplncvrlvol2  40075  dalem19  40142  paddasslem9  40288  pclclN  40351  pclfinN  40360  pclfinclN  40410  pexmidlem8N  40437  trlval3  40647  cdleme22b  40801  cdlemefr29bpre0N  40866  cdlemefr29clN  40867  cdlemefr32fvaN  40869  cdlemefr32fva1  40870  cdlemg31b0N  41154  cdlemg31b0a  41155  cdlemh  41277  dihmeetlem16N  41782  dihmeetlem18N  41784  dihmeetlem19N  41785  dihmeetlem20N  41786  hoidmvlelem1  47041