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Mirrors > Home > ILE Home > Th. List > pm4.72 | GIF version |
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.) |
Ref | Expression |
---|---|
pm4.72 | ⊢ ((φ → ψ) ↔ (ψ ↔ (φ ∨ ψ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olc 631 | . . 3 ⊢ (ψ → (φ ∨ ψ)) | |
2 | pm2.621 665 | . . 3 ⊢ ((φ → ψ) → ((φ ∨ ψ) → ψ)) | |
3 | 1, 2 | impbid2 131 | . 2 ⊢ ((φ → ψ) → (ψ ↔ (φ ∨ ψ))) |
4 | orc 632 | . . 3 ⊢ (φ → (φ ∨ ψ)) | |
5 | bi2 121 | . . 3 ⊢ ((ψ ↔ (φ ∨ ψ)) → ((φ ∨ ψ) → ψ)) | |
6 | 4, 5 | syl5 28 | . 2 ⊢ ((ψ ↔ (φ ∨ ψ)) → (φ → ψ)) |
7 | 3, 6 | impbii 117 | 1 ⊢ ((φ → ψ) ↔ (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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