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

Theorem ccase 931
Description: Inference for combining cases. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Wolf Lammen, 6-Jan-2013.)
Hypotheses
Ref Expression
ccase.1 ((𝜑𝜓) → 𝜏)
ccase.2 ((𝜒𝜓) → 𝜏)
ccase.3 ((𝜑𝜃) → 𝜏)
ccase.4 ((𝜒𝜃) → 𝜏)
Assertion
Ref Expression
ccase (((𝜑𝜒) ∧ (𝜓𝜃)) → 𝜏)

Proof of Theorem ccase
StepHypRef Expression
1 ccase.1 . . 3 ((𝜑𝜓) → 𝜏)
2 ccase.2 . . 3 ((𝜒𝜓) → 𝜏)
31, 2jaoian 767 . 2 (((𝜑𝜒) ∧ 𝜓) → 𝜏)
4 ccase.3 . . 3 ((𝜑𝜃) → 𝜏)
5 ccase.4 . . 3 ((𝜒𝜃) → 𝜏)
64, 5jaoian 767 . 2 (((𝜑𝜒) ∧ 𝜃) → 𝜏)
73, 6jaodan 769 1 (((𝜑𝜒) ∧ (𝜓𝜃)) → 𝜏)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  ccased  932  ccase2  933  undif3ss  3305  nn0gcdsq  11784
  Copyright terms: Public domain W3C validator