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Theorem subadd 7958
Description: Relationship between subtraction and addition. (Contributed by NM, 20-Jan-1997.) (Revised by Mario Carneiro, 21-Dec-2013.)
Assertion
Ref Expression
subadd  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  =  C  <->  ( B  +  C )  =  A ) )

Proof of Theorem subadd
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 subval 7947 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  =  ( iota_ x  e.  CC  ( B  +  x )  =  A ) )
21eqeq1d 2146 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  =  C  <-> 
( iota_ x  e.  CC  ( B  +  x
)  =  A )  =  C ) )
323adant3 1001 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  =  C  <->  ( iota_ x  e.  CC  ( B  +  x )  =  A )  =  C ) )
4 negeu 7946 . . . . 5  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  E! x  e.  CC  ( B  +  x
)  =  A )
5 oveq2 5775 . . . . . . 7  |-  ( x  =  C  ->  ( B  +  x )  =  ( B  +  C ) )
65eqeq1d 2146 . . . . . 6  |-  ( x  =  C  ->  (
( B  +  x
)  =  A  <->  ( B  +  C )  =  A ) )
76riota2 5745 . . . . 5  |-  ( ( C  e.  CC  /\  E! x  e.  CC  ( B  +  x
)  =  A )  ->  ( ( B  +  C )  =  A  <->  ( iota_ x  e.  CC  ( B  +  x )  =  A )  =  C ) )
84, 7sylan2 284 . . . 4  |-  ( ( C  e.  CC  /\  ( B  e.  CC  /\  A  e.  CC ) )  ->  ( ( B  +  C )  =  A  <->  ( iota_ x  e.  CC  ( B  +  x )  =  A )  =  C ) )
983impb 1177 . . 3  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  A  e.  CC )  ->  (
( B  +  C
)  =  A  <->  ( iota_ x  e.  CC  ( B  +  x )  =  A )  =  C ) )
1093com13 1186 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  +  C
)  =  A  <->  ( iota_ x  e.  CC  ( B  +  x )  =  A )  =  C ) )
113, 10bitr4d 190 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  =  C  <->  ( B  +  C )  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   E!wreu 2416   iota_crio 5722  (class class class)co 5767   CCcc 7611    + caddc 7616    - cmin 7926
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-setind 4447  ax-resscn 7705  ax-1cn 7706  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-sub 7928
This theorem is referenced by:  subadd2  7959  subsub23  7960  pncan  7961  pncan3  7963  addsubeq4  7970  subsub2  7983  renegcl  8016  subaddi  8042  subaddd  8084  fzen  9816  nn0ennn  10199  cos2t  11442  cos2tsin  11443  odd2np1  11555  divalgb  11607  sincosq1eq  12905  coskpi  12914
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