Theorem subsub4 7341

Description: Law for double subtraction. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
subsub4	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) - C ) = ( A - ( B + C ) ) )

Proof of Theorem subsub4

Step	Hyp	Ref	Expression
1		nppcan2 7339	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - ( B + C ) ) + C ) = ( A - B ) )
2		simp1 938	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
3		simp2 939	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> B e. CC )
4		subcl 7307	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
5	2, 3, 4	syl2anc 403	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - B ) e. CC )
6		simp3 940	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
7	3, 6	addcld 7138	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B + C ) e. CC )
8		subcl 7307	. . . 4 |- ( ( A e. CC /\ ( B + C ) e. CC ) -> ( A - ( B + C ) ) e. CC )
9	2, 7, 8	syl2anc 403	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - ( B + C ) ) e. CC )
10		subadd2 7312	. . 3 |- ( ( ( A - B ) e. CC /\ C e. CC /\ ( A - ( B + C ) ) e. CC ) -> ( ( ( A - B ) - C ) = ( A - ( B + C ) ) <-> ( ( A - ( B + C ) ) + C ) = ( A - B ) ) )
11	5, 6, 9, 10	syl3anc 1169	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A - B ) - C ) = ( A - ( B + C ) ) <-> ( ( A - ( B + C ) ) + C ) = ( A - B ) ) )
12	1, 11	mpbird 165	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) - C ) = ( A - ( B + C ) ) )

Syntax hints:   -> wi 4   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433  (class class class)co 5532   CCcc 6979   + caddc 6984   - cmin 7279

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-setind 4280  ax-resscn 7068  ax-1cn 7069  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-sub 7281

This theorem is referenced by:  sub32  7342  nnncan  7343  pnpcan  7347  addsub4  7351  subsub4d  7450  2shfti  9719  nn0seqcvgd  10423