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Theorem addlsub 8139
Description: Left-subtraction: Subtraction of the left summand from the result of an addition. (Contributed by BJ, 6-Jun-2019.)
Hypotheses
Ref Expression
addlsub.a  |-  ( ph  ->  A  e.  CC )
addlsub.b  |-  ( ph  ->  B  e.  CC )
addlsub.c  |-  ( ph  ->  C  e.  CC )
Assertion
Ref Expression
addlsub  |-  ( ph  ->  ( ( A  +  B )  =  C  <-> 
A  =  ( C  -  B ) ) )

Proof of Theorem addlsub
StepHypRef Expression
1 oveq1 5781 . . 3  |-  ( ( A  +  B )  =  C  ->  (
( A  +  B
)  -  B )  =  ( C  -  B ) )
2 addlsub.a . . . . 5  |-  ( ph  ->  A  e.  CC )
3 addlsub.b . . . . 5  |-  ( ph  ->  B  e.  CC )
42, 3pncand 8081 . . . 4  |-  ( ph  ->  ( ( A  +  B )  -  B
)  =  A )
5 eqtr2 2158 . . . . . 6  |-  ( ( ( ( A  +  B )  -  B
)  =  ( C  -  B )  /\  ( ( A  +  B )  -  B
)  =  A )  ->  ( C  -  B )  =  A )
65eqcomd 2145 . . . . 5  |-  ( ( ( ( A  +  B )  -  B
)  =  ( C  -  B )  /\  ( ( A  +  B )  -  B
)  =  A )  ->  A  =  ( C  -  B ) )
76a1i 9 . . . 4  |-  ( ph  ->  ( ( ( ( A  +  B )  -  B )  =  ( C  -  B
)  /\  ( ( A  +  B )  -  B )  =  A )  ->  A  =  ( C  -  B
) ) )
84, 7mpan2d 424 . . 3  |-  ( ph  ->  ( ( ( A  +  B )  -  B )  =  ( C  -  B )  ->  A  =  ( C  -  B ) ) )
91, 8syl5 32 . 2  |-  ( ph  ->  ( ( A  +  B )  =  C  ->  A  =  ( C  -  B ) ) )
10 oveq1 5781 . . 3  |-  ( A  =  ( C  -  B )  ->  ( A  +  B )  =  ( ( C  -  B )  +  B ) )
11 addlsub.c . . . . 5  |-  ( ph  ->  C  e.  CC )
1211, 3npcand 8084 . . . 4  |-  ( ph  ->  ( ( C  -  B )  +  B
)  =  C )
13 eqtr 2157 . . . . 5  |-  ( ( ( A  +  B
)  =  ( ( C  -  B )  +  B )  /\  ( ( C  -  B )  +  B
)  =  C )  ->  ( A  +  B )  =  C )
1413a1i 9 . . . 4  |-  ( ph  ->  ( ( ( A  +  B )  =  ( ( C  -  B )  +  B
)  /\  ( ( C  -  B )  +  B )  =  C )  ->  ( A  +  B )  =  C ) )
1512, 14mpan2d 424 . . 3  |-  ( ph  ->  ( ( A  +  B )  =  ( ( C  -  B
)  +  B )  ->  ( A  +  B )  =  C ) )
1610, 15syl5 32 . 2  |-  ( ph  ->  ( A  =  ( C  -  B )  ->  ( A  +  B )  =  C ) )
179, 16impbid 128 1  |-  ( ph  ->  ( ( A  +  B )  =  C  <-> 
A  =  ( C  -  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480  (class class class)co 5774   CCcc 7625    + caddc 7630    - cmin 7940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452  ax-resscn 7719  ax-1cn 7720  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-addcom 7727  ax-addass 7729  ax-distr 7731  ax-i2m1 7732  ax-0id 7735  ax-rnegex 7736  ax-cnre 7738
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-sub 7942
This theorem is referenced by:  addrsub  8140  subexsub  8141  nn0ob  11611  oddennn  11911
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