Theorem subsub2 8203

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 8174	. . . . 5 |- ( ( B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
2	1	3adant1 1017	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
3		simp1 999	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
4		simp3 1001	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
5		simp2 1000	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> B e. CC )
6		subcl 8174	. . . . 5 |- ( ( C e. CC /\ B e. CC ) -> ( C - B ) e. CC )
7	4, 5, 6	syl2anc 411	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C - B ) e. CC )
8	2, 3, 7	add12d 8142	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B - C ) + ( A + ( C - B ) ) ) = ( A + ( ( B - C ) + ( C - B ) ) ) )
9		npncan2 8202	. . . . 5 |- ( ( B e. CC /\ C e. CC ) -> ( ( B - C ) + ( C - B ) ) = 0 )
10	9	3adant1 1017	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B - C ) + ( C - B ) ) = 0 )
11	10	oveq2d 5907	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( ( B - C ) + ( C - B ) ) ) = ( A + 0 ) )
12	3	addid1d 8124	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + 0 ) = A )
13	8, 11, 12	3eqtrd 2226	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B - C ) + ( A + ( C - B ) ) ) = A )
14	3, 7	addcld 7995	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( C - B ) ) e. CC )
15		subadd 8178	. . 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 1249	. 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 167	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - ( B - C ) ) = ( A + ( C - B ) ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2160  (class class class)co 5891   CCcc 7827   0cc0 7829   + caddc 7832   - cmin 8146

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-setind 4551  ax-resscn 7921  ax-1cn 7922  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-addcom 7929  ax-addass 7931  ax-distr 7933  ax-i2m1 7934  ax-0id 7937  ax-rnegex 7938  ax-cnre 7940

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-iota 5193  df-fun 5233  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-sub 8148

This theorem is referenced by:  nncan  8204  subsub  8205  subsub3  8207  ppncan  8217  subadd4  8219  subsub2d  8315