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Theorem subadd 7214
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 7203 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  =  ( iota_ x  e.  CC  ( B  +  x )  =  A ) )
21eqeq1d 2048 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  =  C  <-> 
( iota_ x  e.  CC  ( B  +  x
)  =  A )  =  C ) )
323adant3 924 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  =  C  <->  ( iota_ x  e.  CC  ( B  +  x )  =  A )  =  C ) )
4 negeu 7202 . . . . 5  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  E! x  e.  CC  ( B  +  x
)  =  A )
5 oveq2 5520 . . . . . . 7  |-  ( x  =  C  ->  ( B  +  x )  =  ( B  +  C ) )
65eqeq1d 2048 . . . . . 6  |-  ( x  =  C  ->  (
( B  +  x
)  =  A  <->  ( B  +  C )  =  A ) )
76riota2 5490 . . . . 5  |-  ( ( C  e.  CC  /\  E! x  e.  CC  ( B  +  x
)  =  A )  ->  ( ( B  +  C )  =  A  <->  ( iota_ x  e.  CC  ( B  +  x )  =  A )  =  C ) )
84, 7sylan2 270 . . . 4  |-  ( ( C  e.  CC  /\  ( B  e.  CC  /\  A  e.  CC ) )  ->  ( ( B  +  C )  =  A  <->  ( iota_ x  e.  CC  ( B  +  x )  =  A )  =  C ) )
983impb 1100 . . 3  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  A  e.  CC )  ->  (
( B  +  C
)  =  A  <->  ( iota_ x  e.  CC  ( B  +  x )  =  A )  =  C ) )
1093com13 1109 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  +  C
)  =  A  <->  ( iota_ x  e.  CC  ( B  +  x )  =  A )  =  C ) )
113, 10bitr4d 180 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 97    <-> wb 98    /\ w3a 885    = wceq 1243    e. wcel 1393   E!wreu 2308   iota_crio 5467  (class class class)co 5512   CCcc 6887    + caddc 6892    - cmin 7182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262  ax-resscn 6976  ax-1cn 6977  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-sub 7184
This theorem is referenced by:  subadd2  7215  subsub23  7216  pncan  7217  pncan3  7219  addsubeq4  7226  subsub2  7239  renegcl  7272  subaddi  7298  subaddd  7340  fzen  8907  nn0ennn  9209
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