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| Mirrors > Home > MPE Home > Th. List > ifpfal | Structured version Visualization version GIF version | ||
| Description: Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 4534. This is essentially dedlemb 1047. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 25-Jun-2020.) |
| Ref | Expression |
|---|---|
| ifpfal | ⊢ (¬ 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜒)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ifpn 1074 | . 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓)) | |
| 2 | ifptru 1075 | . 2 ⊢ (¬ 𝜑 → (if-(¬ 𝜑, 𝜒, 𝜓) ↔ 𝜒)) | |
| 3 | 1, 2 | bitrid 283 | 1 ⊢ (¬ 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜒)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 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: ifpid 1077 elimh 1083 axprlem3 5425 axpr 5427 axprlem3OLD 5428 axprlem5OLD 5430 wlkdlem4 29703 lfgriswlk 29706 2pthnloop 29751 eupth2lem3lem4 30250 satfv1lem 35367 wl-3xorfal 37473 sn-axprlem3 42256 |
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