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Theorem subadd 9700
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 9688 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  =  ( iota_ x  e.  CC  ( B  +  x )  =  A ) )
21eqeq1d 2452 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  =  C  <-> 
( iota_ x  e.  CC  ( B  +  x
)  =  A )  =  C ) )
323adant3 1008 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  =  C  <->  ( iota_ x  e.  CC  ( B  +  x )  =  A )  =  C ) )
4 negeu 9687 . . . . 5  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  E! x  e.  CC  ( B  +  x
)  =  A )
5 oveq2 6184 . . . . . . 7  |-  ( x  =  C  ->  ( B  +  x )  =  ( B  +  C ) )
65eqeq1d 2452 . . . . . 6  |-  ( x  =  C  ->  (
( B  +  x
)  =  A  <->  ( B  +  C )  =  A ) )
76riota2 6160 . . . . 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 1184 . . 3  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  A  e.  CC )  ->  (
( B  +  C
)  =  A  <->  ( iota_ x  e.  CC  ( B  +  x )  =  A )  =  C ) )
1093com13 1193 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  +  C
)  =  A  <->  ( iota_ x  e.  CC  ( B  +  x )  =  A )  =  C ) )
113, 10bitr4d 256 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 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757   E!wreu 2794   iota_crio 6136  (class class class)co 6176   CCcc 9367    + caddc 9372    - cmin 9682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-po 4725  df-so 4726  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-er 7187  df-en 7397  df-dom 7398  df-sdom 7399  df-pnf 9507  df-mnf 9508  df-ltxr 9510  df-sub 9684
This theorem is referenced by:  subadd2  9701  subsub23  9702  pncan  9703  pncan3  9705  addsubeq4  9712  subsub2  9724  renegcli  9757  subaddi  9782  subaddd  9824  fzen  11554  nn0ennn  11888  hashssdif  12255  cos2t  13550  cos2tsin  13551  odd2np1  13680  divalglem4  13688  divalglem8  13692  divalgb  13696  mplmonmul  17636  sincosq1eq  22076  coskpi  22084  sto2i  25762  tan2h  28548  fdc  28765
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