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Mirrors > Home > ILE Home > Th. List > xnegeq | Unicode version |
Description: Equality of two extended numbers with in front of them. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xnegeq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2124 | . . 3 | |
2 | eqeq1 2124 | . . . 4 | |
3 | negeq 7923 | . . . 4 | |
4 | 2, 3 | ifbieq2d 3466 | . . 3 |
5 | 1, 4 | ifbieq2d 3466 | . 2 |
6 | df-xneg 9527 | . 2 | |
7 | df-xneg 9527 | . 2 | |
8 | 5, 6, 7 | 3eqtr4g 2175 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1316 cif 3444 cpnf 7765 cmnf 7766 cneg 7902 cxne 9524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rex 2399 df-rab 2402 df-v 2662 df-un 3045 df-if 3445 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-iota 5058 df-fv 5101 df-ov 5745 df-neg 7904 df-xneg 9527 |
This theorem is referenced by: xnegcl 9583 xnegneg 9584 xneg11 9585 xltnegi 9586 xnegid 9610 xnegdi 9619 xsubge0 9632 xposdif 9633 xlesubadd 9634 xrnegiso 10999 infxrnegsupex 11000 xrminmax 11002 xrminrecl 11010 xrminadd 11012 xblss2ps 12500 xblss2 12501 |
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