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Theorem subaddeqd 7901
Description: Transfer two terms of a subtraction to an addition in an equality. (Contributed by Thierry Arnoux, 2-Feb-2020.)
Hypotheses
Ref Expression
subaddeqd.a  |-  ( ph  ->  A  e.  CC )
subaddeqd.b  |-  ( ph  ->  B  e.  CC )
subaddeqd.c  |-  ( ph  ->  C  e.  CC )
subaddeqd.d  |-  ( ph  ->  D  e.  CC )
subaddeqd.1  |-  ( ph  ->  ( A  +  B
)  =  ( C  +  D ) )
Assertion
Ref Expression
subaddeqd  |-  ( ph  ->  ( A  -  D
)  =  ( C  -  B ) )

Proof of Theorem subaddeqd
StepHypRef Expression
1 subaddeqd.1 . . . 4  |-  ( ph  ->  ( A  +  B
)  =  ( C  +  D ) )
21oveq1d 5681 . . 3  |-  ( ph  ->  ( ( A  +  B )  -  ( D  +  B )
)  =  ( ( C  +  D )  -  ( D  +  B ) ) )
3 subaddeqd.c . . . . 5  |-  ( ph  ->  C  e.  CC )
4 subaddeqd.d . . . . 5  |-  ( ph  ->  D  e.  CC )
53, 4addcomd 7687 . . . 4  |-  ( ph  ->  ( C  +  D
)  =  ( D  +  C ) )
65oveq1d 5681 . . 3  |-  ( ph  ->  ( ( C  +  D )  -  ( D  +  B )
)  =  ( ( D  +  C )  -  ( D  +  B ) ) )
72, 6eqtrd 2121 . 2  |-  ( ph  ->  ( ( A  +  B )  -  ( D  +  B )
)  =  ( ( D  +  C )  -  ( D  +  B ) ) )
8 subaddeqd.a . . 3  |-  ( ph  ->  A  e.  CC )
9 subaddeqd.b . . 3  |-  ( ph  ->  B  e.  CC )
108, 4, 9pnpcan2d 7885 . 2  |-  ( ph  ->  ( ( A  +  B )  -  ( D  +  B )
)  =  ( A  -  D ) )
114, 3, 9pnpcand 7884 . 2  |-  ( ph  ->  ( ( D  +  C )  -  ( D  +  B )
)  =  ( C  -  B ) )
127, 10, 113eqtr3d 2129 1  |-  ( ph  ->  ( A  -  D
)  =  ( C  -  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1290    e. wcel 1439  (class class class)co 5666   CCcc 7402    + caddc 7407    - cmin 7707
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-setind 4366  ax-resscn 7491  ax-1cn 7492  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-addcom 7499  ax-addass 7501  ax-distr 7503  ax-i2m1 7504  ax-0id 7507  ax-rnegex 7508  ax-cnre 7510
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-iota 4993  df-fun 5030  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-sub 7709
This theorem is referenced by: (None)
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