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Theorem ccase 984
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 820 . 2 (((𝜑𝜒) ∧ 𝜓) → 𝜏)
4 ccase.3 . . 3 ((𝜑𝜃) → 𝜏)
5 ccase.4 . . 3 ((𝜒𝜃) → 𝜏)
64, 5jaoian 820 . 2 (((𝜑𝜒) ∧ 𝜃) → 𝜏)
73, 6jaodan 822 1 (((𝜑𝜒) ∧ (𝜓𝜃)) → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383
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 196  df-or 384  df-an 385
This theorem is referenced by:  ccased  985  ccase2  986  undif3OLD  3848  ssprsseq  4297  injresinjlem  12450  prodmo  14505  nn0gcdsq  15298  symgextf1  17664  cnmsgnsubg  19742  kelac2lem  36652
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