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Theorem addlsub 8100
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 5749 . . 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 8042 . . . 4  |-  ( ph  ->  ( ( A  +  B )  -  B
)  =  A )
5 eqtr2 2136 . . . . . 6  |-  ( ( ( ( A  +  B )  -  B
)  =  ( C  -  B )  /\  ( ( A  +  B )  -  B
)  =  A )  ->  ( C  -  B )  =  A )
65eqcomd 2123 . . . . 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 5749 . . 3  |-  ( A  =  ( C  -  B )  ->  ( A  +  B )  =  ( ( C  -  B )  +  B ) )
11 addlsub.c . . . . 5  |-  ( ph  ->  C  e.  CC )
1211, 3npcand 8045 . . . 4  |-  ( ph  ->  ( ( C  -  B )  +  B
)  =  C )
13 eqtr 2135 . . . . 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 1316    e. wcel 1465  (class class class)co 5742   CCcc 7586    + caddc 7591    - cmin 7901
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-setind 4422  ax-resscn 7680  ax-1cn 7681  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-sub 7903
This theorem is referenced by:  addrsub  8101  subexsub  8102  nn0ob  11532  oddennn  11832
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