New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  pm5.74 GIF version

Theorem pm5.74 235
 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 178 . . . 4 ((ψχ) → (ψχ))
21imim3i 55 . . 3 ((φ → (ψχ)) → ((φψ) → (φχ)))
3 bi2 189 . . . 4 ((ψχ) → (χψ))
43imim3i 55 . . 3 ((φ → (ψχ)) → ((φχ) → (φψ)))
52, 4impbid 183 . 2 ((φ → (ψχ)) → ((φψ) ↔ (φχ)))
6 bi1 178 . . . 4 (((φψ) ↔ (φχ)) → ((φψ) → (φχ)))
76pm2.86d 93 . . 3 (((φψ) ↔ (φχ)) → (φ → (ψχ)))
8 bi2 189 . . . 4 (((φψ) ↔ (φχ)) → ((φχ) → (φψ)))
98pm2.86d 93 . . 3 (((φψ) ↔ (φχ)) → (φ → (χψ)))
107, 9impbidd 181 . 2 (((φψ) ↔ (φχ)) → (φ → (ψχ)))
115, 10impbii 180 1 ((φ → (ψχ)) ↔ ((φψ) ↔ (φχ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176 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 177 This theorem is referenced by:  pm5.74i  236  pm5.74ri  237  pm5.74d  238  pm5.74rd  239  bibi2d  309  pm5.32  617  orbidi  898
 Copyright terms: Public domain W3C validator