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Mirrors > Home > ILE Home > Th. List > xleadd1 | Unicode version |
Description: Weakened version of xleadd1a 9656 under which the reverse implication is true. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xleadd1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 7811 | . . 3 | |
2 | xleadd1a 9656 | . . . 4 | |
3 | 2 | ex 114 | . . 3 |
4 | 1, 3 | syl3an3 1251 | . 2 |
5 | simp1 981 | . . . . 5 | |
6 | 1 | 3ad2ant3 1004 | . . . . 5 |
7 | xaddcl 9643 | . . . . 5 | |
8 | 5, 6, 7 | syl2anc 408 | . . . 4 |
9 | simp2 982 | . . . . 5 | |
10 | xaddcl 9643 | . . . . 5 | |
11 | 9, 6, 10 | syl2anc 408 | . . . 4 |
12 | xnegcl 9615 | . . . . 5 | |
13 | 6, 12 | syl 14 | . . . 4 |
14 | xleadd1a 9656 | . . . . 5 | |
15 | 14 | ex 114 | . . . 4 |
16 | 8, 11, 13, 15 | syl3anc 1216 | . . 3 |
17 | xpncan 9654 | . . . . 5 | |
18 | 17 | 3adant2 1000 | . . . 4 |
19 | xpncan 9654 | . . . . 5 | |
20 | 19 | 3adant1 999 | . . . 4 |
21 | 18, 20 | breq12d 3942 | . . 3 |
22 | 16, 21 | sylibd 148 | . 2 |
23 | 4, 22 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 w3a 962 wceq 1331 wcel 1480 class class class wbr 3929 (class class class)co 5774 cr 7619 cxr 7799 cle 7801 cxne 9556 cxad 9557 |
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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-apti 7735 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-xneg 9559 df-xadd 9560 |
This theorem is referenced by: xsubge0 9664 xlesubadd 9666 |
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