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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 4472. This is essentially dedlema 1040. (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 363 | . 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓))) | |
2 | orc 863 | . . . 4 ⊢ (𝜑 → (𝜑 ∨ 𝜒)) | |
3 | 2 | biantrud 534 | . . 3 ⊢ (𝜑 → ((𝜑 → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒)))) |
4 | dfifp3 1060 | . . 3 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒))) | |
5 | 3, 4 | syl6bbr 291 | . 2 ⊢ (𝜑 → ((𝜑 → 𝜓) ↔ if-(𝜑, 𝜓, 𝜒))) |
6 | 1, 5 | bitr2d 282 | 1 ⊢ (𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 if-wif 1057 |
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 209 df-an 399 df-or 844 df-ifp 1058 |
This theorem is referenced by: ifpfal 1069 ifpid 1070 elimh 1076 elimhOLD 1077 dedt 1078 dedtOLD 1079 axprlem3 5317 axprlem4 5318 wlkl1loop 27413 lfgrwlkprop 27463 eupth2lem3lem3 28003 satfv1lem 32604 sn-axprlem3 39102 |
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