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Theorem ccased 1025
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 1024 . 2 (((𝜓𝜃) ∧ (𝜒𝜏)) → (𝜑𝜂))
109com12 32 1 (𝜑 → (((𝜓𝜃) ∧ (𝜒𝜏)) → 𝜂))
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 197  df-or 384  df-an 385
This theorem is referenced by:  fpwwe2lem13  9676  mulge0  10758  zmulcl  11638  gcdabs  15472  lcmabs  15540  pospo  17194  mulgass  17800  indistopon  21027  lgsdir2lem5  25274  outsideofeq  32564  smprngopr  34182  cdlemg33  36519  monotoddzzfi  38027  acongtr  38065
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