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Theorem subaddeqd 8339
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 5903 . . 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 8121 . . . 4  |-  ( ph  ->  ( C  +  D
)  =  ( D  +  C ) )
65oveq1d 5903 . . 3  |-  ( ph  ->  ( ( C  +  D )  -  ( D  +  B )
)  =  ( ( D  +  C )  -  ( D  +  B ) ) )
72, 6eqtrd 2220 . 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 8319 . 2  |-  ( ph  ->  ( ( A  +  B )  -  ( D  +  B )
)  =  ( A  -  D ) )
114, 3, 9pnpcand 8318 . 2  |-  ( ph  ->  ( ( D  +  C )  -  ( D  +  B )
)  =  ( C  -  B ) )
127, 10, 113eqtr3d 2228 1  |-  ( ph  ->  ( A  -  D
)  =  ( C  -  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1363    e. wcel 2158  (class class class)co 5888   CCcc 7822    + caddc 7827    - cmin 8141
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 7916  ax-1cn 7917  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-distr 7928  ax-i2m1 7929  ax-0id 7932  ax-rnegex 7933  ax-cnre 7935
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 8143
This theorem is referenced by: (None)
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