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Theorem subcan2 8172
Description: Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
Assertion
Ref Expression
subcan2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
)  =  ( B  -  C )  <->  A  =  B ) )

Proof of Theorem subcan2
StepHypRef Expression
1 simp1 997 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
2 simp3 999 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
3 subcl 8146 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C
)  e.  CC )
433adant1 1015 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C )  e.  CC )
5 subadd2 8151 . . 3  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  ( B  -  C )  e.  CC )  ->  (
( A  -  C
)  =  ( B  -  C )  <->  ( ( B  -  C )  +  C )  =  A ) )
61, 2, 4, 5syl3anc 1238 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
)  =  ( B  -  C )  <->  ( ( B  -  C )  +  C )  =  A ) )
7 npcan 8156 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( B  -  C )  +  C
)  =  B )
873adant1 1015 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  -  C
)  +  C )  =  B )
98eqeq1d 2186 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( B  -  C )  +  C
)  =  A  <->  B  =  A ) )
10 eqcom 2179 . . 3  |-  ( B  =  A  <->  A  =  B )
119, 10bitrdi 196 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( B  -  C )  +  C
)  =  A  <->  A  =  B ) )
126, 11bitrd 188 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
)  =  ( B  -  C )  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148  (class class class)co 5869   CCcc 7800    + caddc 7805    - cmin 8118
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-setind 4533  ax-resscn 7894  ax-1cn 7895  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-sub 8120
This theorem is referenced by:  subeq0  8173  subcan2i  8241  subcan2d  8300  subcan2ad  8303  zextlt  9334
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