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Theorem 2addsub 9320
Description: Law for subtraction and addition. (Contributed by NM, 20-Nov-2005.)
Assertion
Ref Expression
2addsub  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( ( A  +  B )  +  C )  -  D
)  =  ( ( ( A  +  C
)  -  D )  +  B ) )

Proof of Theorem 2addsub
StepHypRef Expression
1 add32 9281 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  +  C )  =  ( ( A  +  C )  +  B ) )
213expa 1154 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  C  e.  CC )  ->  ( ( A  +  B )  +  C )  =  ( ( A  +  C
)  +  B ) )
32adantrr 699 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  B )  +  C
)  =  ( ( A  +  C )  +  B ) )
43oveq1d 6097 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( ( A  +  B )  +  C )  -  D
)  =  ( ( ( A  +  C
)  +  B )  -  D ) )
5 addcl 9073 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  +  C
)  e.  CC )
6 addsub 9317 . . . . 5  |-  ( ( ( A  +  C
)  e.  CC  /\  B  e.  CC  /\  D  e.  CC )  ->  (
( ( A  +  C )  +  B
)  -  D )  =  ( ( ( A  +  C )  -  D )  +  B ) )
763expb 1155 . . . 4  |-  ( ( ( A  +  C
)  e.  CC  /\  ( B  e.  CC  /\  D  e.  CC ) )  ->  ( (
( A  +  C
)  +  B )  -  D )  =  ( ( ( A  +  C )  -  D )  +  B
) )
85, 7sylan 459 . . 3  |-  ( ( ( A  e.  CC  /\  C  e.  CC )  /\  ( B  e.  CC  /\  D  e.  CC ) )  -> 
( ( ( A  +  C )  +  B )  -  D
)  =  ( ( ( A  +  C
)  -  D )  +  B ) )
98an4s 801 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( ( A  +  C )  +  B )  -  D
)  =  ( ( ( A  +  C
)  -  D )  +  B ) )
104, 9eqtrd 2469 1  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( ( A  +  B )  +  C )  -  D
)  =  ( ( ( A  +  C
)  -  D )  +  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726  (class class class)co 6082   CCcc 8989    + caddc 8994    - cmin 9292
This theorem is referenced by:  2addsubd  9462
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-po 4504  df-so 4505  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-riota 6550  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-ltxr 9126  df-sub 9294
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