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Theorem subsub2 8335
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 8306 . . . . 5 |- ( ( B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
213adant1 1018 . . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
3 simp1 1000 . . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
4 simp3 1002 . . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
5 simp2 1001 . . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> B e. CC )
6 subcl 8306 . . . . 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 8274 . . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B - C ) + ( A + ( C - B ) ) ) = ( A + ( ( B - C ) + ( C - B ) ) ) )
9 npncan2 8334 . . . . 5 |- ( ( B e. CC /\ C e. CC ) -> ( ( B - C ) + ( C - B ) ) = 0 )
1093adant1 1018 . . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B - C ) + ( C - B ) ) = 0 )
1110oveq2d 5983 . . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( ( B - C ) + ( C - B ) ) ) = ( A + 0 ) )
123addridd 8256 . . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + 0 ) = A )
138, 11, 123eqtrd 2244 . 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B - C ) + ( A + ( C - B ) ) ) = A )
143, 7addcld 8127 . . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( C - B ) ) e. CC )
15 subadd 8310 . . 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 1250 . 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 981   = wceq 1373   e. wcel 2178  (class class class)co 5967   CCcc 7958   0cc0 7960   + caddc 7963   - cmin 8278
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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-setind 4603  ax-resscn 8052  ax-1cn 8053  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-sub 8280
This theorem is referenced by:  nncan  8336  subsub  8337  subsub3  8339  ppncan  8349  subadd4  8351  subsub2d  8447
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