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Mirrors > Home > ILE Home > Th. List > pncan3 | Unicode version |
Description: Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.) |
Ref | Expression |
---|---|
pncan3 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2187 |
. 2
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2 | simpr 110 |
. . 3
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3 | simpl 109 |
. . 3
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4 | subcl 8170 |
. . . 4
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5 | 4 | ancoms 268 |
. . 3
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6 | subadd 8174 |
. . 3
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7 | 2, 3, 5, 6 | syl3anc 1248 |
. 2
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8 | 1, 7 | mpbii 148 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-setind 4548 ax-resscn 7917 ax-1cn 7918 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-addcom 7925 ax-addass 7927 ax-distr 7929 ax-i2m1 7930 ax-0id 7933 ax-rnegex 7934 ax-cnre 7936 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-sub 8144 |
This theorem is referenced by: npcan 8180 nncan 8200 npncan3 8209 negid 8218 pncan3i 8248 pncan3d 8285 subdi 8356 posdif 8426 fzonmapblen 10201 frecfzen2 10441 bernneq2 10656 hashfz 10815 isumshft 11512 dvdssubr 11860 dvef 14484 sincosq2sgn 14544 sincosq3sgn 14545 sincosq4sgn 14546 logdivlti 14598 |
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