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Theorem pm4.87 547
 Description: Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Eric Schmidt, 26-Oct-2006.)
Assertion
Ref Expression
pm4.87 (((((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))) ∧ ((𝜑 → (𝜓𝜒)) ↔ (𝜓 → (𝜑𝜒)))) ∧ ((𝜓 → (𝜑𝜒)) ↔ ((𝜓𝜑) → 𝜒)))

Proof of Theorem pm4.87
StepHypRef Expression
1 impexp 261 . . 3 (((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒)))
2 bi2.04 247 . . 3 ((𝜑 → (𝜓𝜒)) ↔ (𝜓 → (𝜑𝜒)))
31, 2pm3.2i 270 . 2 ((((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))) ∧ ((𝜑 → (𝜓𝜒)) ↔ (𝜓 → (𝜑𝜒))))
4 impexp 261 . . 3 (((𝜓𝜑) → 𝜒) ↔ (𝜓 → (𝜑𝜒)))
54bicomi 131 . 2 ((𝜓 → (𝜑𝜒)) ↔ ((𝜓𝜑) → 𝜒))
63, 5pm3.2i 270 1 (((((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))) ∧ ((𝜑 → (𝜓𝜒)) ↔ (𝜓 → (𝜑𝜒)))) ∧ ((𝜓 → (𝜑𝜒)) ↔ ((𝜓𝜑) → 𝜒)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107 This theorem depends on definitions:  df-bi 116 This theorem is referenced by: (None)
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