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

Theorem ccased 1034
 Description: Deduction for combining cases. (Contributed by NM, 9-May-2004.)
Hypotheses
Ref Expression
ccased.1 (𝜑 → ((𝜓𝜒) → 𝜂))
ccased.2 (𝜑 → ((𝜃𝜒) → 𝜂))
ccased.3 (𝜑 → ((𝜓𝜏) → 𝜂))
ccased.4 (𝜑 → ((𝜃𝜏) → 𝜂))
Assertion
Ref Expression
ccased (𝜑 → (((𝜓𝜃) ∧ (𝜒𝜏)) → 𝜂))

Proof of Theorem ccased
StepHypRef Expression
1 ccased.1 . . . 4 (𝜑 → ((𝜓𝜒) → 𝜂))
21com12 32 . . 3 ((𝜓𝜒) → (𝜑𝜂))
3 ccased.2 . . . 4 (𝜑 → ((𝜃𝜒) → 𝜂))
43com12 32 . . 3 ((𝜃𝜒) → (𝜑𝜂))
5 ccased.3 . . . 4 (𝜑 → ((𝜓𝜏) → 𝜂))
65com12 32 . . 3 ((𝜓𝜏) → (𝜑𝜂))
7 ccased.4 . . . 4 (𝜑 → ((𝜃𝜏) → 𝜂))
87com12 32 . . 3 ((𝜃𝜏) → (𝜑𝜂))
92, 4, 6, 8ccase 1033 . 2 (((𝜓𝜃) ∧ (𝜒𝜏)) → (𝜑𝜂))
109com12 32 1 (𝜑 → (((𝜓𝜃) ∧ (𝜒𝜏)) → 𝜂))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844 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 210  df-an 400  df-or 845 This theorem is referenced by:  fpwwe2lem13  10041  mulge0  11135  zmulcl  12009  gcdabs  15854  lcmabs  15926  pospo  17562  mulgass  18243  indistopon  21585  lgsdir2lem5  25892  outsideofeq  33599  smprngopr  35372  cdlemg33  37889  monotoddzzfi  39690  acongtr  39726
 Copyright terms: Public domain W3C validator