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Theorem pm4.72 813
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 701 . . 3 (𝜓 → (𝜑𝜓))
2 pm2.621 737 . . 3 ((𝜑𝜓) → ((𝜑𝜓) → 𝜓))
31, 2impbid2 142 . 2 ((𝜑𝜓) → (𝜓 ↔ (𝜑𝜓)))
4 orc 702 . . 3 (𝜑 → (𝜑𝜓))
5 bi2 129 . . 3 ((𝜓 ↔ (𝜑𝜓)) → ((𝜑𝜓) → 𝜓))
64, 5syl5 32 . 2 ((𝜓 ↔ (𝜑𝜓)) → (𝜑𝜓))
73, 6impbii 125 1 ((𝜑𝜓) ↔ (𝜓 ↔ (𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wo 698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  bigolden  940  ssequn1  3249
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