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Mirrors > Home > MPE Home > Th. List > ifpid | Structured version Visualization version GIF version |
Description: Value of the conditional operator for propositions when the same proposition is returned in either case. Analogue for propositions of ifid 4569. This is essentially pm4.42 1053. (Contributed by BJ, 20-Sep-2019.) |
Ref | Expression |
---|---|
ifpid | ⊢ (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifptru 1075 | . 2 ⊢ (𝜑 → (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓)) | |
2 | ifpfal 1076 | . 2 ⊢ (¬ 𝜑 → (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓)) | |
3 | 1, 2 | pm2.61i 182 | 1 ⊢ (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 if-wif 1062 |
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 df-an 398 df-or 847 df-ifp 1063 |
This theorem is referenced by: wl-1mintru2 36370 |
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