Theorem subsub4 8522

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 8520	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - ( B + C ) ) + C ) = ( A - B ) )
2		simp1 1024	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
3		simp2 1025	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> B e. CC )
4		subcl 8488	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
5	2, 3, 4	syl2anc 411	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - B ) e. CC )
6		simp3 1026	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
7	3, 6	addcld 8309	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B + C ) e. CC )
8		subcl 8488	. . . 4 |- ( ( A e. CC /\ ( B + C ) e. CC ) -> ( A - ( B + C ) ) e. CC )
9	2, 7, 8	syl2anc 411	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - ( B + C ) ) e. CC )
10		subadd2 8493	. . 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 1274	. 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 167	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) - C ) = ( A - ( B + C ) ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205  (class class class)co 6058   CCcc 8141   + caddc 8146   - cmin 8460

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 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-setind 4664  ax-resscn 8235  ax-1cn 8236  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-sub 8462

This theorem is referenced by:  sub32  8523  nnncan  8524  pnpcan  8528  addsub4  8532  subsub4d  8631  2shfti  11541  nn0seqcvgd  12763