Theorem subadd 8360

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.

Step	Hyp	Ref	Expression
1		subval 8349	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) = ( iota_ x e. CC ( B + x ) = A ) )
2	1	eqeq1d 2238	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) = C <-> ( iota_ x e. CC ( B + x ) = A ) = C ) )
3	2	3adant3 1041	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) = C <-> ( iota_ x e. CC ( B + x ) = A ) = C ) )
4		negeu 8348	. . . . 5 |- ( ( B e. CC /\ A e. CC ) -> E! x e. CC ( B + x ) = A )
5		oveq2 6015	. . . . . . 7 |- ( x = C -> ( B + x ) = ( B + C ) )
6	5	eqeq1d 2238	. . . . . 6 |- ( x = C -> ( ( B + x ) = A <-> ( B + C ) = A ) )
7	6	riota2 5984	. . . . 5 |- ( ( C e. CC /\ E! x e. CC ( B + x ) = A ) -> ( ( B + C ) = A <-> ( iota_ x e. CC ( B + x ) = A ) = C ) )
8	4, 7	sylan2 286	. . . 4 |- ( ( C e. CC /\ ( B e. CC /\ A e. CC ) ) -> ( ( B + C ) = A <-> ( iota_ x e. CC ( B + x ) = A ) = C ) )
9	8	3impb 1223	. . 3 |- ( ( C e. CC /\ B e. CC /\ A e. CC ) -> ( ( B + C ) = A <-> ( iota_ x e. CC ( B + x ) = A ) = C ) )
10	9	3com13 1232	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B + C ) = A <-> ( iota_ x e. CC ( B + x ) = A ) = C ) )
11	3, 10	bitr4d 191	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) = C <-> ( B + C ) = A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   E!wreu 2510   iota_crio 5959  (class class class)co 6007   CCcc 8008   + caddc 8013   - cmin 8328

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-setind 4629  ax-resscn 8102  ax-1cn 8103  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-distr 8114  ax-i2m1 8115  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-sub 8330

This theorem is referenced by:  subadd2  8361  subsub23  8362  pncan  8363  pncan3  8365  addsubeq4  8372  subsub2  8385  renegcl  8418  subaddi  8444  subaddd  8486  fzen  10251  nn0ennn  10667  cos2t  12276  cos2tsin  12277  odd2np1  12399  divalgb  12451  sincosq1eq  15528  coskpi  15537