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Theorem subaddeqd 8345
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 5906 . . 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 8127 . . . 4  |-  ( ph  ->  ( C  +  D
)  =  ( D  +  C ) )
65oveq1d 5906 . . 3  |-  ( ph  ->  ( ( C  +  D )  -  ( D  +  B )
)  =  ( ( D  +  C )  -  ( D  +  B ) ) )
72, 6eqtrd 2222 . 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 8325 . 2  |-  ( ph  ->  ( ( A  +  B )  -  ( D  +  B )
)  =  ( A  -  D ) )
114, 3, 9pnpcand 8324 . 2  |-  ( ph  ->  ( ( D  +  C )  -  ( D  +  B )
)  =  ( C  -  B ) )
127, 10, 113eqtr3d 2230 1  |-  ( ph  ->  ( A  -  D
)  =  ( C  -  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2160  (class class class)co 5891   CCcc 7828    + caddc 7833    - cmin 8147
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-setind 4551  ax-resscn 7922  ax-1cn 7923  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-addcom 7930  ax-addass 7932  ax-distr 7934  ax-i2m1 7935  ax-0id 7938  ax-rnegex 7939  ax-cnre 7941
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-iota 5193  df-fun 5233  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-sub 8149
This theorem is referenced by: (None)
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