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Mirrors > Home > MPE Home > Th. List > impbi | Structured version Visualization version GIF version |
Description: Property of the biconditional connective. (Contributed by NM, 11-May-1999.) |
Ref | Expression |
---|---|
impbi | ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bi 206 | . . 3 ⊢ ¬ (((𝜑 ↔ 𝜓) → ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑))) → ¬ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓))) | |
2 | simprim 166 | . . 3 ⊢ (¬ (((𝜑 ↔ 𝜓) → ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑))) → ¬ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓))) → (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓)) |
4 | 3 | expi 165 | 1 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 |
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 206 |
This theorem is referenced by: impbii 208 impbidd 209 impbid21d 210 dfbi1 212 impimprbi 826 bj-bisym 35932 bj-moeub 36192 eqsbc2VD 44064 orbi1rVD 44072 3impexpVD 44080 3impexpbicomVD 44081 imbi12VD 44097 sbcim2gVD 44099 sb5ALTVD 44137 |
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