Theorem subcan 8214

Description: Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Assertion

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

Proof of Theorem subcan

Step	Hyp	Ref	Expression
1		simp2 998	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> B e. CC )
2		simp1 997	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
3	1, 2	addcomd 8110	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B + A ) = ( A + B ) )
4	3	eqeq1d 2186	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B + A ) = ( A + C ) <-> ( A + B ) = ( A + C ) ) )
5		simp3 999	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
6		addsubeq4 8174	. . 3 |- ( ( ( B e. CC /\ A e. CC ) /\ ( A e. CC /\ C e. CC ) ) -> ( ( B + A ) = ( A + C ) <-> ( A - B ) = ( A - C ) ) )
7	1, 2, 2, 5, 6	syl22anc 1239	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B + A ) = ( A + C ) <-> ( A - B ) = ( A - C ) ) )
8		addcan 8139	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) <-> B = C ) )
9	4, 7, 8	3bitr3d 218	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) = ( A - C ) <-> B = C ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148  (class class class)co 5877   CCcc 7811   + caddc 7816   - cmin 8130

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-setind 4538  ax-resscn 7905  ax-1cn 7906  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-sub 8132

This theorem is referenced by:  subcani  8252  subcand  8311  subcanad  8313