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Theorem pm4.72 938
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 ↔ (𝜑𝜓)))

Proof of Theorem pm4.72
StepHypRef Expression
1 olc 398 . . 3 (𝜓 → (𝜑𝜓))
2 pm2.621 423 . . 3 ((𝜑𝜓) → ((𝜑𝜓) → 𝜓))
31, 2impbid2 216 . 2 ((𝜑𝜓) → (𝜓 ↔ (𝜑𝜓)))
4 orc 399 . . 3 (𝜑 → (𝜑𝜓))
5 biimpr 210 . . 3 ((𝜓 ↔ (𝜑𝜓)) → ((𝜑𝜓) → 𝜓))
64, 5syl5 34 . 2 ((𝜓 ↔ (𝜑𝜓)) → (𝜑𝜓))
73, 6impbii 199 1 ((𝜑𝜓) ↔ (𝜓 ↔ (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382
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  df-or 384
This theorem is referenced by:  bigolden  996  cadan  1588  ssequn1  3816  ssunsn2  4391  vtxd0nedgb  26440  bj-consensusALT  32688  elpaddn0  35404
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