MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  syl231anc Structured version   Visualization version   GIF version

Theorem syl231anc 1382
Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
Hypotheses
Ref Expression
syl3anc.1 (𝜑𝜓)
syl3anc.2 (𝜑𝜒)
syl3anc.3 (𝜑𝜃)
syl3Xanc.4 (𝜑𝜏)
syl23anc.5 (𝜑𝜂)
syl33anc.6 (𝜑𝜁)
syl231anc.7 (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ 𝜁) → 𝜎)
Assertion
Ref Expression
syl231anc (𝜑𝜎)

Proof of Theorem syl231anc
StepHypRef Expression
1 syl3anc.1 . . 3 (𝜑𝜓)
2 syl3anc.2 . . 3 (𝜑𝜒)
31, 2jca 512 . 2 (𝜑 → (𝜓𝜒))
4 syl3anc.3 . 2 (𝜑𝜃)
5 syl3Xanc.4 . 2 (𝜑𝜏)
6 syl23anc.5 . 2 (𝜑𝜂)
7 syl33anc.6 . 2 (𝜑𝜁)
8 syl231anc.7 . 2 (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ 𝜁) → 𝜎)
93, 4, 5, 6, 7, 8syl131anc 1375 1 (𝜑𝜎)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079
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 208  df-an 397  df-3an 1081
This theorem is referenced by:  syl232anc  1389  isosctr  25326  axeuclid  26676  dalawlem3  36889  dalawlem6  36892  cdlemd7  37220  cdleme18c  37309  cdlemi  37836  cdlemk7  37864  cdlemk11  37865  cdlemk7u  37886  cdlemk11u  37887  cdlemk19xlem  37958  cdlemk55u1  37981  cdlemk56  37987
  Copyright terms: Public domain W3C validator