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Theorem subadd 9861
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 9849 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  =  ( iota_ x  e.  CC  ( B  +  x )  =  A ) )
21eqeq1d 2406 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  =  C  <-> 
( iota_ x  e.  CC  ( B  +  x
)  =  A )  =  C ) )
323adant3 1019 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  =  C  <->  ( iota_ x  e.  CC  ( B  +  x )  =  A )  =  C ) )
4 negeu 9848 . . . . 5  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  E! x  e.  CC  ( B  +  x
)  =  A )
5 oveq2 6288 . . . . . . 7  |-  ( x  =  C  ->  ( B  +  x )  =  ( B  +  C ) )
65eqeq1d 2406 . . . . . 6  |-  ( x  =  C  ->  (
( B  +  x
)  =  A  <->  ( B  +  C )  =  A ) )
76riota2 6264 . . . . 5  |-  ( ( C  e.  CC  /\  E! x  e.  CC  ( B  +  x
)  =  A )  ->  ( ( B  +  C )  =  A  <->  ( iota_ x  e.  CC  ( B  +  x )  =  A )  =  C ) )
84, 7sylan2 474 . . . 4  |-  ( ( C  e.  CC  /\  ( B  e.  CC  /\  A  e.  CC ) )  ->  ( ( B  +  C )  =  A  <->  ( iota_ x  e.  CC  ( B  +  x )  =  A )  =  C ) )
983impb 1195 . . 3  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  A  e.  CC )  ->  (
( B  +  C
)  =  A  <->  ( iota_ x  e.  CC  ( B  +  x )  =  A )  =  C ) )
1093com13 1204 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  +  C
)  =  A  <->  ( iota_ x  e.  CC  ( B  +  x )  =  A )  =  C ) )
113, 10bitr4d 258 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  =  C  <->  ( B  +  C )  =  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844   E!wreu 2758   iota_crio 6241  (class class class)co 6280   CCcc 9522    + caddc 9527    - cmin 9843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-po 4746  df-so 4747  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9662  df-mnf 9663  df-ltxr 9665  df-sub 9845
This theorem is referenced by:  subadd2  9862  subsub23  9863  pncan  9864  pncan3  9866  addsubeq4  9873  subsub2  9885  renegcli  9918  subaddi  9945  subaddd  9987  fzen  11759  nn0ennn  12132  hashssdif  12526  cos2t  14124  cos2tsin  14125  odd2np1  14257  divalglem4  14265  divalglem8  14269  divalgb  14273  mplmonmul  18448  sincosq1eq  23199  coskpi  23207  sto2i  27582  tan2h  31432  fdc  31533
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