ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  syl33anc GIF version

Theorem syl33anc 1231
Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
Hypotheses
Ref Expression
sylXanc.1 (𝜑𝜓)
sylXanc.2 (𝜑𝜒)
sylXanc.3 (𝜑𝜃)
sylXanc.4 (𝜑𝜏)
sylXanc.5 (𝜑𝜂)
sylXanc.6 (𝜑𝜁)
syl33anc.7 (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁)) → 𝜎)
Assertion
Ref Expression
syl33anc (𝜑𝜎)

Proof of Theorem syl33anc
StepHypRef Expression
1 sylXanc.1 . . 3 (𝜑𝜓)
2 sylXanc.2 . . 3 (𝜑𝜒)
3 sylXanc.3 . . 3 (𝜑𝜃)
41, 2, 33jca 1161 . 2 (𝜑 → (𝜓𝜒𝜃))
5 sylXanc.4 . 2 (𝜑𝜏)
6 sylXanc.5 . 2 (𝜑𝜂)
7 sylXanc.6 . 2 (𝜑𝜁)
8 syl33anc.7 . 2 (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁)) → 𝜎)
94, 5, 6, 7, 8syl13anc 1218 1 (𝜑𝜎)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116  df-3an 964
This theorem is referenced by:  strleund  12073  iscnp4  12413  cnpnei  12414  cnptopco  12417  cncnp  12425  cnptopresti  12433  lmtopcnp  12445  txcnp  12466  xmetrtri  12571  bl2in  12598  blhalf  12603  blssps  12622  blss  12623
  Copyright terms: Public domain W3C validator