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Theorem sub32 7398
Description: Swap the second and third terms in a double subtraction. (Contributed by NM, 19-Aug-2005.)
Assertion
Ref Expression
sub32  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  -  C )  =  ( ( A  -  C )  -  B ) )

Proof of Theorem sub32
StepHypRef Expression
1 addcom 7301 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C
)  =  ( C  +  B ) )
213adant1 957 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C )  =  ( C  +  B ) )
32oveq2d 5553 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  ( B  +  C ) )  =  ( A  -  ( C  +  B )
) )
4 subsub4 7397 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  -  C )  =  ( A  -  ( B  +  C
) ) )
5 subsub4 7397 . . 3  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  B  e.  CC )  ->  (
( A  -  C
)  -  B )  =  ( A  -  ( C  +  B
) ) )
653com23 1145 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
)  -  B )  =  ( A  -  ( C  +  B
) ) )
73, 4, 63eqtr4d 2124 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    /\ w3a 920    = wceq 1285    e. wcel 1434  (class class class)co 5537   CCcc 7030    + caddc 7035    - cmin 7335
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-setind 4282  ax-resscn 7119  ax-1cn 7120  ax-icn 7122  ax-addcl 7123  ax-addrcl 7124  ax-mulcl 7125  ax-addcom 7127  ax-addass 7129  ax-distr 7131  ax-i2m1 7132  ax-0id 7135  ax-rnegex 7136  ax-cnre 7138
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-iota 4891  df-fun 4928  df-fv 4934  df-riota 5493  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-sub 7337
This theorem is referenced by:  nnncan1  7400  nnncan2  7401  sub32d  7507  2shfti  9846
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