Theorem subcan2 8382

Description: Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)

Assertion

Ref	Expression
subcan2	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - C ) = ( B - C ) <-> A = B ) )

Proof of Theorem subcan2

Step	Hyp	Ref	Expression
1		simp1 1021	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
2		simp3 1023	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
3		subcl 8356	. . . 4 |- ( ( B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
4	3	3adant1 1039	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
5		subadd2 8361	. . 3 |- ( ( A e. CC /\ C e. CC /\ ( B - C ) e. CC ) -> ( ( A - C ) = ( B - C ) <-> ( ( B - C ) + C ) = A ) )
6	1, 2, 4, 5	syl3anc 1271	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - C ) = ( B - C ) <-> ( ( B - C ) + C ) = A ) )
7		npcan 8366	. . . . 5 |- ( ( B e. CC /\ C e. CC ) -> ( ( B - C ) + C ) = B )
8	7	3adant1 1039	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B - C ) + C ) = B )
9	8	eqeq1d 2238	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( B - C ) + C ) = A <-> B = A ) )
10		eqcom 2231	. . 3 |- ( B = A <-> A = B )
11	9, 10	bitrdi 196	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( B - C ) + C ) = A <-> A = B ) )
12	6, 11	bitrd 188	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - C ) = ( B - C ) <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200  (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:  subeq0  8383  subcan2i  8451  subcan2d  8510  subcan2ad  8513  zextlt  9550