Theorem subadd 8476

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 8465	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) = ( iota_ x e. CC ( B + x ) = A ) )
2	1	eqeq1d 2241	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) = C <-> ( iota_ x e. CC ( B + x ) = A ) = C ) )
3	2	3adant3 1044	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) = C <-> ( iota_ x e. CC ( B + x ) = A ) = C ) )
4		negeu 8464	. . . . 5 |- ( ( B e. CC /\ A e. CC ) -> E! x e. CC ( B + x ) = A )
5		oveq2 6058	. . . . . . 7 |- ( x = C -> ( B + x ) = ( B + C ) )
6	5	eqeq1d 2241	. . . . . 6 |- ( x = C -> ( ( B + x ) = A <-> ( B + C ) = A ) )
7	6	riota2 6027	. . . . 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 1226	. . 3 |- ( ( C e. CC /\ B e. CC /\ A e. CC ) -> ( ( B + C ) = A <-> ( iota_ x e. CC ( B + x ) = A ) = C ) )
10	9	3com13 1235	. 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 1005   = wceq 1398   e. wcel 2203   E!wreu 2522   iota_crio 6002  (class class class)co 6050   CCcc 8125   + caddc 8130   - cmin 8444

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-setind 4659  ax-resscn 8219  ax-1cn 8220  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-sub 8446

This theorem is referenced by:  subadd2  8477  subsub23  8478  pncan  8479  pncan3  8481  addsubeq4  8488  subsub2  8501  renegcl  8534  subaddi  8560  subaddd  8602  fzen  10377  nn0ennn  10795  cos2t  12436  cos2tsin  12437  odd2np1  12559  divalgb  12611  sincosq1eq  15704  coskpi  15713