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Mirrors > Home > ILE Home > Th. List > subaddeqd | Unicode version |
Description: Transfer two terms of a subtraction to an addition in an equality. (Contributed by Thierry Arnoux, 2-Feb-2020.) |
Ref | Expression |
---|---|
subaddeqd.a |
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subaddeqd.b |
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subaddeqd.c |
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subaddeqd.d |
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subaddeqd.1 |
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Ref | Expression |
---|---|
subaddeqd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subaddeqd.1 |
. . . 4
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2 | 1 | oveq1d 5829 |
. . 3
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3 | subaddeqd.c |
. . . . 5
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4 | subaddeqd.d |
. . . . 5
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5 | 3, 4 | addcomd 8005 |
. . . 4
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6 | 5 | oveq1d 5829 |
. . 3
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7 | 2, 6 | eqtrd 2187 |
. 2
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8 | subaddeqd.a |
. . 3
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9 | subaddeqd.b |
. . 3
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10 | 8, 4, 9 | pnpcan2d 8203 |
. 2
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11 | 4, 3, 9 | pnpcand 8202 |
. 2
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12 | 7, 10, 11 | 3eqtr3d 2195 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-14 2128 ax-ext 2136 ax-sep 4078 ax-pow 4130 ax-pr 4164 ax-setind 4490 ax-resscn 7803 ax-1cn 7804 ax-icn 7806 ax-addcl 7807 ax-addrcl 7808 ax-mulcl 7809 ax-addcom 7811 ax-addass 7813 ax-distr 7815 ax-i2m1 7816 ax-0id 7819 ax-rnegex 7820 ax-cnre 7822 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-ral 2437 df-rex 2438 df-reu 2439 df-rab 2441 df-v 2711 df-sbc 2934 df-dif 3100 df-un 3102 df-in 3104 df-ss 3111 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-uni 3769 df-br 3962 df-opab 4022 df-id 4248 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-iota 5128 df-fun 5165 df-fv 5171 df-riota 5770 df-ov 5817 df-oprab 5818 df-mpo 5819 df-sub 8027 |
This theorem is referenced by: (None) |
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