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Theorem subsub2 8397
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
StepHypRef Expression
1 subcl 8368 . . . . 5 |- ( ( B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
213adant1 1039 . . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
3 simp1 1021 . . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
4 simp3 1023 . . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
5 simp2 1022 . . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> B e. CC )
6 subcl 8368 . . . . 5 |- ( ( C e. CC /\ B e. CC ) -> ( C - B ) e. CC )
74, 5, 6syl2anc 411 . . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C - B ) e. CC )
82, 3, 7add12d 8336 . . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B - C ) + ( A + ( C - B ) ) ) = ( A + ( ( B - C ) + ( C - B ) ) ) )
9 npncan2 8396 . . . . 5 |- ( ( B e. CC /\ C e. CC ) -> ( ( B - C ) + ( C - B ) ) = 0 )
1093adant1 1039 . . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B - C ) + ( C - B ) ) = 0 )
1110oveq2d 6029 . . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( ( B - C ) + ( C - B ) ) ) = ( A + 0 ) )
123addridd 8318 . . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + 0 ) = A )
138, 11, 123eqtrd 2266 . 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B - C ) + ( A + ( C - B ) ) ) = A )
143, 7addcld 8189 . . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( C - B ) ) e. CC )
15 subadd 8372 . . 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 ) )
163, 2, 14, 15syl3anc 1271 . 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - ( B - C ) ) = ( A + ( C - B ) ) <-> ( ( B - C ) + ( A + ( C - B ) ) ) = A ) )
1713, 16mpbird 167 1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - ( B - C ) ) = ( A + ( C - B ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200  (class class class)co 6013   CCcc 8020   0cc0 8022   + caddc 8025   - cmin 8340
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-setind 4633  ax-resscn 8114  ax-1cn 8115  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133
This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-sub 8342
This theorem is referenced by:  nncan  8398  subsub  8399  subsub3  8401  ppncan  8411  subadd4  8413  subsub2d  8509
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