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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 2144 | . . 3 | |
2 | eqeq1 2144 | . . . 4 | |
3 | negeq 7948 | . . . 4 | |
4 | 2, 3 | ifbieq2d 3491 | . . 3 |
5 | 1, 4 | ifbieq2d 3491 | . 2 |
6 | df-xneg 9552 | . 2 | |
7 | df-xneg 9552 | . 2 | |
8 | 5, 6, 7 | 3eqtr4g 2195 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 cif 3469 cpnf 7790 cmnf 7791 cneg 7927 cxne 9549 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rex 2420 df-rab 2423 df-v 2683 df-un 3070 df-if 3470 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-iota 5083 df-fv 5126 df-ov 5770 df-neg 7929 df-xneg 9552 |
This theorem is referenced by: xnegcl 9608 xnegneg 9609 xneg11 9610 xltnegi 9611 xnegid 9635 xnegdi 9644 xsubge0 9657 xposdif 9658 xlesubadd 9659 xrnegiso 11024 infxrnegsupex 11025 xrminmax 11027 xrminrecl 11035 xrminadd 11037 xblss2ps 12562 xblss2 12563 |
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