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Theorem pm5.74 259
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 biimp 205 . . . 4 ((𝜓𝜒) → (𝜓𝜒))
21imim3i 64 . . 3 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
3 biimpr 210 . . . 4 ((𝜓𝜒) → (𝜒𝜓))
43imim3i 64 . . 3 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜒) → (𝜑𝜓)))
52, 4impbid 202 . 2 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) ↔ (𝜑𝜒)))
6 biimp 205 . . . 4 (((𝜑𝜓) ↔ (𝜑𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
76pm2.86d 107 . . 3 (((𝜑𝜓) ↔ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))
8 biimpr 210 . . . 4 (((𝜑𝜓) ↔ (𝜑𝜒)) → ((𝜑𝜒) → (𝜑𝜓)))
98pm2.86d 107 . . 3 (((𝜑𝜓) ↔ (𝜑𝜒)) → (𝜑 → (𝜒𝜓)))
107, 9impbidd 200 . 2 (((𝜑𝜓) ↔ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))
115, 10impbii 199 1 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196
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
This theorem is referenced by:  pm5.74i  260  pm5.74ri  261  pm5.74d  262  pm5.74rd  263  bibi2d  331  pm5.32  669  orbidi  993
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