Theorem subadd 8531

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 8520	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) = ( iota_ x e. CC ( B + x ) = A ) )
2	1	eqeq1d 2247	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) = C <-> ( iota_ x e. CC ( B + x ) = A ) = C ) )
3	2	3adant3 1048	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) = C <-> ( iota_ x e. CC ( B + x ) = A ) = C ) )
4		negeu 8519	. . . . 5 |- ( ( B e. CC /\ A e. CC ) -> E! x e. CC ( B + x ) = A )
5		oveq2 6093	. . . . . . 7 |- ( x = C -> ( B + x ) = ( B + C ) )
6	5	eqeq1d 2247	. . . . . 6 |- ( x = C -> ( ( B + x ) = A <-> ( B + C ) = A ) )
7	6	riota2 6062	. . . . 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 1230	. . 3 |- ( ( C e. CC /\ B e. CC /\ A e. CC ) -> ( ( B + C ) = A <-> ( iota_ x e. CC ( B + x ) = A ) = C ) )
10	9	3com13 1239	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   E!wreu 2530   iota_crio 6037  (class class class)co 6085   CCcc 8178   + caddc 8183   - cmin 8499

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-setind 4684  ax-resscn 8272  ax-1cn 8273  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-addcom 8280  ax-addass 8282  ax-distr 8284  ax-i2m1 8285  ax-0id 8288  ax-rnegex 8289  ax-cnre 8291

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-sub 8501

This theorem is used by:  subadd2  8532  subsub23  8533  pncan  8534  pncan3  8536  addsubeq4  8543  subsub2  8556  renegcl  8589  subaddi  8615  subaddd  8657  fzen  10458  nn0ennn  10885  cos2t  12536  cos2tsin  12537  odd2np1  12659  divalgb  12711  sincosq1eq  16032  coskpi  16041