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Theorem ccased 872
 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 27 . . 3 ((𝜓𝜒) → (𝜑𝜂))
3 ccased.2 . . . 4 (𝜑 → ((𝜃𝜒) → 𝜂))
43com12 27 . . 3 ((𝜃𝜒) → (𝜑𝜂))
5 ccased.3 . . . 4 (𝜑 → ((𝜓𝜏) → 𝜂))
65com12 27 . . 3 ((𝜓𝜏) → (𝜑𝜂))
7 ccased.4 . . . 4 (𝜑 → ((𝜃𝜏) → 𝜂))
87com12 27 . . 3 ((𝜃𝜏) → (𝜑𝜂))
92, 4, 6, 8ccase 871 . 2 (((𝜓𝜃) ∧ (𝜒𝜏)) → (𝜑𝜂))
109com12 27 1 (𝜑 → (((𝜓𝜃) ∧ (𝜒𝜏)) → 𝜂))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 629 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  zmulcl  8297
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