New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  pm4.72 GIF version

Theorem pm4.72 846
 Description: Implication in terms of biconditional and disjunction. Theorem *4.72 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Jan-2013.)
Assertion
Ref Expression
pm4.72 ((φψ) ↔ (ψ ↔ (φ ψ)))

Proof of Theorem pm4.72
StepHypRef Expression
1 olc 373 . . 3 (ψ → (φ ψ))
2 pm2.621 397 . . 3 ((φψ) → ((φ ψ) → ψ))
31, 2impbid2 195 . 2 ((φψ) → (ψ ↔ (φ ψ)))
4 orc 374 . . 3 (φ → (φ ψ))
5 bi2 189 . . 3 ((ψ ↔ (φ ψ)) → ((φ ψ) → ψ))
64, 5syl5 28 . 2 ((ψ ↔ (φ ψ)) → (φψ))
73, 6impbii 180 1 ((φψ) ↔ (ψ ↔ (φ ψ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8 This theorem depends on definitions:  df-bi 177  df-or 359 This theorem is referenced by:  bigolden  901  ssequn1  3433  ssunsn2  3865
 Copyright terms: Public domain W3C validator