Theorem subsub2 8126

Description: Law for double subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
subsub2	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - ( B - C ) ) = ( A + ( C - B ) ) )

Proof of Theorem subsub2

Step	Hyp	Ref	Expression
1		subcl 8097	. . . . 5 |- ( ( B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
2	1	3adant1 1005	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
3		simp1 987	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
4		simp3 989	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
5		simp2 988	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> B e. CC )
6		subcl 8097	. . . . 5 |- ( ( C e. CC /\ B e. CC ) -> ( C - B ) e. CC )
7	4, 5, 6	syl2anc 409	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C - B ) e. CC )
8	2, 3, 7	add12d 8065	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B - C ) + ( A + ( C - B ) ) ) = ( A + ( ( B - C ) + ( C - B ) ) ) )
9		npncan2 8125	. . . . 5 |- ( ( B e. CC /\ C e. CC ) -> ( ( B - C ) + ( C - B ) ) = 0 )
10	9	3adant1 1005	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B - C ) + ( C - B ) ) = 0 )
11	10	oveq2d 5858	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( ( B - C ) + ( C - B ) ) ) = ( A + 0 ) )
12	3	addid1d 8047	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + 0 ) = A )
13	8, 11, 12	3eqtrd 2202	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B - C ) + ( A + ( C - B ) ) ) = A )
14	3, 7	addcld 7918	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( C - B ) ) e. CC )
15		subadd 8101	. . 3 |- ( ( A e. CC /\ ( B - C ) e. CC /\ ( A + ( C - B ) ) e. CC ) -> ( ( A - ( B - C ) ) = ( A + ( C - B ) ) <-> ( ( B - C ) + ( A + ( C - B ) ) ) = A ) )
16	3, 2, 14, 15	syl3anc 1228	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - ( B - C ) ) = ( A + ( C - B ) ) <-> ( ( B - C ) + ( A + ( C - B ) ) ) = A ) )
17	13, 16	mpbird 166	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - ( B - C ) ) = ( A + ( C - B ) ) )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136  (class class class)co 5842   CCcc 7751   0cc0 7753   + caddc 7756   - cmin 8069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514  ax-resscn 7845  ax-1cn 7846  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-sub 8071

This theorem is referenced by:  nncan  8127  subsub  8128  subsub3  8130  ppncan  8140  subadd4  8142  subsub2d  8238