Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  pm5.74 Structured version   Visualization version   GIF version

Theorem pm5.74 258
 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 204 . . . 4 ((𝜓𝜒) → (𝜓𝜒))
21imim3i 62 . . 3 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
3 biimpr 209 . . . 4 ((𝜓𝜒) → (𝜒𝜓))
43imim3i 62 . . 3 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜒) → (𝜑𝜓)))
52, 4impbid 201 . 2 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) ↔ (𝜑𝜒)))
6 biimp 204 . . . 4 (((𝜑𝜓) ↔ (𝜑𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
76pm2.86d 105 . . 3 (((𝜑𝜓) ↔ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))
8 biimpr 209 . . . 4 (((𝜑𝜓) ↔ (𝜑𝜒)) → ((𝜑𝜒) → (𝜑𝜓)))
98pm2.86d 105 . . 3 (((𝜑𝜓) ↔ (𝜑𝜒)) → (𝜑 → (𝜒𝜓)))
107, 9impbidd 199 . 2 (((𝜑𝜓) ↔ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))
115, 10impbii 198 1 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195 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 196 This theorem is referenced by:  pm5.74i  259  pm5.74ri  260  pm5.74d  261  pm5.74rd  262  bibi2d  331  pm5.32  666  orbidi  969
 Copyright terms: Public domain W3C validator