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Theorem addlsub 7593
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 5570 . . 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 7539 . . . 4  |-  ( ph  ->  ( ( A  +  B )  -  B
)  =  A )
5 eqtr2 2101 . . . . . 6  |-  ( ( ( ( A  +  B )  -  B
)  =  ( C  -  B )  /\  ( ( A  +  B )  -  B
)  =  A )  ->  ( C  -  B )  =  A )
65eqcomd 2088 . . . . 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 419 . . 3  |-  ( ph  ->  ( ( ( A  +  B )  -  B )  =  ( C  -  B )  ->  A  =  ( C  -  B ) ) )
91, 8syl5 32 . 2  |-  ( ph  ->  ( ( A  +  B )  =  C  ->  A  =  ( C  -  B ) ) )
10 oveq1 5570 . . 3  |-  ( A  =  ( C  -  B )  ->  ( A  +  B )  =  ( ( C  -  B )  +  B ) )
11 addlsub.c . . . . 5  |-  ( ph  ->  C  e.  CC )
1211, 3npcand 7542 . . . 4  |-  ( ph  ->  ( ( C  -  B )  +  B
)  =  C )
13 eqtr 2100 . . . . 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 419 . . 3  |-  ( ph  ->  ( ( A  +  B )  =  ( ( C  -  B
)  +  B )  ->  ( A  +  B )  =  C ) )
1610, 15syl5 32 . 2  |-  ( ph  ->  ( A  =  ( C  -  B )  ->  ( A  +  B )  =  C ) )
179, 16impbid 127 1  |-  ( ph  ->  ( ( A  +  B )  =  C  <-> 
A  =  ( C  -  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434  (class class class)co 5563   CCcc 7093    + caddc 7098    - cmin 7398
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 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-setind 4308  ax-resscn 7182  ax-1cn 7183  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-sub 7400
This theorem is referenced by:  addrsub  7594  subexsub  7595  nn0ob  10515  oddennn  10812
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