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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 4448. 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 1076 | . 2 ⊢ (¬ 𝜑 → (if-(¬ 𝜑, 𝜒, 𝜓) ↔ 𝜒)) | |
3 | 1, 2 | syl5bb 286 | 1 ⊢ (¬ 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜒)) |
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: ifpid 1078 elimh 1085 axprlem3 5318 axprlem5 5320 wlkdlem4 27773 lfgriswlk 27776 2pthnloop 27818 eupth2lem3lem4 28314 satfv1lem 33037 wl-3xorfal 35380 sn-axprlem3 39908 |
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