Theorem subadd 8381

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 8370	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) = ( iota_ x e. CC ( B + x ) = A ) )
2	1	eqeq1d 2240	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) = C <-> ( iota_ x e. CC ( B + x ) = A ) = C ) )
3	2	3adant3 1043	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) = C <-> ( iota_ x e. CC ( B + x ) = A ) = C ) )
4		negeu 8369	. . . . 5 |- ( ( B e. CC /\ A e. CC ) -> E! x e. CC ( B + x ) = A )
5		oveq2 6025	. . . . . . 7 |- ( x = C -> ( B + x ) = ( B + C ) )
6	5	eqeq1d 2240	. . . . . 6 |- ( x = C -> ( ( B + x ) = A <-> ( B + C ) = A ) )
7	6	riota2 5994	. . . . 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 1225	. . 3 |- ( ( C e. CC /\ B e. CC /\ A e. CC ) -> ( ( B + C ) = A <-> ( iota_ x e. CC ( B + x ) = A ) = C ) )
10	9	3com13 1234	. 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 1004   = wceq 1397   e. wcel 2202   E!wreu 2512   iota_crio 5969  (class class class)co 6017   CCcc 8029   + caddc 8034   - cmin 8349

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-setind 4635  ax-resscn 8123  ax-1cn 8124  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-sub 8351

This theorem is referenced by:  subadd2  8382  subsub23  8383  pncan  8384  pncan3  8386  addsubeq4  8393  subsub2  8406  renegcl  8439  subaddi  8465  subaddd  8507  fzen  10277  nn0ennn  10694  cos2t  12310  cos2tsin  12311  odd2np1  12433  divalgb  12485  sincosq1eq  15562  coskpi  15571