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Mirrors > Home > MPE Home > Th. List > Mathboxes > ifpnim2 | Structured version Visualization version GIF version |
Description: Restate negated implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.) |
Ref | Expression |
---|---|
ifpnim2 | ⊢ (¬ (𝜑 → 𝜓) ↔ if-(𝜓, ¬ 𝜓, 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifpnot23c 40776 | . 2 ⊢ (¬ if-(𝜓, 𝜓, ¬ 𝜑) ↔ if-(𝜓, ¬ 𝜓, 𝜑)) | |
2 | ifpim4 40790 | . 2 ⊢ ((𝜑 → 𝜓) ↔ if-(𝜓, 𝜓, ¬ 𝜑)) | |
3 | 1, 2 | xchnxbir 336 | 1 ⊢ (¬ (𝜑 → 𝜓) ↔ if-(𝜓, ¬ 𝜓, 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 if-wif 1063 |
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 210 df-an 400 df-or 848 df-ifp 1064 |
This theorem is referenced by: (None) |
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