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| Mirrors > Home > MPE Home > Th. List > ifptru | Structured version Visualization version GIF version | ||
| Description: Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of iftrue 4531. This is essentially dedlema 1046. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 10-Jul-2020.) |
| Ref | Expression |
|---|---|
| ifptru | ⊢ (𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | biimt 360 | . 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓))) | |
| 2 | orc 868 | . . . 4 ⊢ (𝜑 → (𝜑 ∨ 𝜒)) | |
| 3 | 2 | biantrud 531 | . . 3 ⊢ (𝜑 → ((𝜑 → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒)))) |
| 4 | dfifp3 1066 | . . 3 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒))) | |
| 5 | 3, 4 | bitr4di 289 | . 2 ⊢ (𝜑 → ((𝜑 → 𝜓) ↔ if-(𝜑, 𝜓, 𝜒))) |
| 6 | 1, 5 | bitr2d 280 | 1 ⊢ (𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 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 207 df-an 396 df-or 849 df-ifp 1064 |
| This theorem is referenced by: ifpfal 1076 ifpid 1077 elimh 1083 dedt 1084 axprlem3 5425 axpr 5427 axprlem3OLD 5428 axprlem4OLD 5429 wlkl1loop 29656 lfgrwlkprop 29705 eupth2lem3lem3 30249 satfv1lem 35367 wl-3xortru 37472 sn-axprlem3 42256 |
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