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Theorem subaddeqd 8099
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 5757 . . 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 7881 . . . 4  |-  ( ph  ->  ( C  +  D
)  =  ( D  +  C ) )
65oveq1d 5757 . . 3  |-  ( ph  ->  ( ( C  +  D )  -  ( D  +  B )
)  =  ( ( D  +  C )  -  ( D  +  B ) ) )
72, 6eqtrd 2150 . 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 8079 . 2  |-  ( ph  ->  ( ( A  +  B )  -  ( D  +  B )
)  =  ( A  -  D ) )
114, 3, 9pnpcand 8078 . 2  |-  ( ph  ->  ( ( D  +  C )  -  ( D  +  B )
)  =  ( C  -  B ) )
127, 10, 113eqtr3d 2158 1  |-  ( ph  ->  ( A  -  D
)  =  ( C  -  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465  (class class class)co 5742   CCcc 7586    + caddc 7591    - cmin 7901
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-setind 4422  ax-resscn 7680  ax-1cn 7681  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-sub 7903
This theorem is referenced by: (None)
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