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| Mirrors > Home > MPE Home > Th. List > iffalsei | Structured version Visualization version GIF version | ||
| Description: Inference associated with iffalse 4534. (Contributed by BJ, 7-Oct-2018.) |
| Ref | Expression |
|---|---|
| iffalsei.1 | ⊢ ¬ 𝜑 |
| Ref | Expression |
|---|---|
| iffalsei | ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iffalsei.1 | . 2 ⊢ ¬ 𝜑 | |
| 2 | iffalse 4534 | . 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ifcif 4525 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-if 4526 |
| This theorem is referenced by: ssttrcl 9755 ttrclselem2 9766 sum0 15757 prod0 15979 prmo4 17165 prmo6 17167 itg0 25815 vieta1lem2 26353 right1s 27934 vtxval0 29056 iedgval0 29057 ex-prmo 30478 dfrdg2 35796 dfrdg4 35952 fwddifnp1 36166 bj-pr21val 37014 bj-pr22val 37020 imsqrtvalex 43659 clsk1indlem4 44057 clsk1indlem1 44058 refsum2cnlem1 45042 limsup10ex 45788 iblempty 45980 fouriersw 46246 |
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