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Theorem ccase 1033
 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 954 . 2 (((𝜑𝜒) ∧ 𝜓) → 𝜏)
4 ccase.3 . . 3 ((𝜑𝜃) → 𝜏)
5 ccase.4 . . 3 ((𝜒𝜃) → 𝜏)
64, 5jaoian 954 . 2 (((𝜑𝜒) ∧ 𝜃) → 𝜏)
73, 6jaodan 955 1 (((𝜑𝜒) ∧ (𝜓𝜃)) → 𝜏)
 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:  ccased  1034  ccase2  1035  ssprsseq  4731  prel12g  4767  injresinjlem  13152  prodmo  15281  nn0gcdsq  16081  symgextf1  18540  cnmsgnsubg  20264  nn0rppwr  39434  nn0expgcd  39436  kelac2lem  39938
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