Theorem subsub2 8449

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 8420	. . . . 5 |- ( ( B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
2	1	3adant1 1042	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
3		simp1 1024	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
4		simp3 1026	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
5		simp2 1025	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> B e. CC )
6		subcl 8420	. . . . 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 8388	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B - C ) + ( A + ( C - B ) ) ) = ( A + ( ( B - C ) + ( C - B ) ) ) )
9		npncan2 8448	. . . . 5 |- ( ( B e. CC /\ C e. CC ) -> ( ( B - C ) + ( C - B ) ) = 0 )
10	9	3adant1 1042	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B - C ) + ( C - B ) ) = 0 )
11	10	oveq2d 6044	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( ( B - C ) + ( C - B ) ) ) = ( A + 0 ) )
12	3	addridd 8370	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + 0 ) = A )
13	8, 11, 12	3eqtrd 2268	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B - C ) + ( A + ( C - B ) ) ) = A )
14	3, 7	addcld 8241	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( C - B ) ) e. CC )
15		subadd 8424	. . 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 1274	. 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 1005   = wceq 1398   e. wcel 2202  (class class class)co 6028   CCcc 8073   0cc0 8075   + caddc 8078   - cmin 8392

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-setind 4641  ax-resscn 8167  ax-1cn 8168  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-sub 8394

This theorem is referenced by:  nncan  8450  subsub  8451  subsub3  8453  ppncan  8463  subadd4  8465  subsub2d  8561