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Theorem pm4.72 735
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 631 . . 3 (ψ → (φ ψ))
2 pm2.621 665 . . 3 ((φψ) → ((φ ψ) → ψ))
31, 2impbid2 131 . 2 ((φψ) → (ψ ↔ (φ ψ)))
4 orc 632 . . 3 (φ → (φ ψ))
5 bi2 121 . . 3 ((ψ ↔ (φ ψ)) → ((φ ψ) → ψ))
64, 5syl5 28 . 2 ((ψ ↔ (φ ψ)) → (φψ))
73, 6impbii 117 1 ((φψ) ↔ (ψ ↔ (φ ψ)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   wo 628
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  bigolden  861  ssequn1  3107
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