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Theorem pm5.74 168
 Description: Distribution of implication over biconditional. Theorem *5.74 of [WhiteheadRussell] p. 126. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 11-Apr-2013.)
Assertion
Ref Expression
pm5.74 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))

Proof of Theorem pm5.74
StepHypRef Expression
1 bi1 111 . . . 4 ((𝜓𝜒) → (𝜓𝜒))
21imim3i 55 . . 3 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
3 bi2 121 . . . 4 ((𝜓𝜒) → (𝜒𝜓))
43imim3i 55 . . 3 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜒) → (𝜑𝜓)))
52, 4impbid 120 . 2 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) ↔ (𝜑𝜒)))
6 bi1 111 . . . 4 (((𝜑𝜓) ↔ (𝜑𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
76pm2.86d 93 . . 3 (((𝜑𝜓) ↔ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))
8 bi2 121 . . . 4 (((𝜑𝜓) ↔ (𝜑𝜒)) → ((𝜑𝜒) → (𝜑𝜓)))
98pm2.86d 93 . . 3 (((𝜑𝜓) ↔ (𝜑𝜒)) → (𝜑 → (𝜒𝜓)))
107, 9impbidd 118 . 2 (((𝜑𝜓) ↔ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))
115, 10impbii 117 1 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  pm5.74i  169  pm5.74ri  170  pm5.74d  171  pm5.74rd  172  bibi2d  221
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