ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  subeqxfrd Unicode version

Theorem subeqxfrd 8384
Description: Transfer two terms of a subtraction in an equality. (Contributed by Thierry Arnoux, 2-Feb-2020.)
Hypotheses
Ref Expression
subeqxfrd.a  |-  ( ph  ->  A  e.  CC )
subeqxfrd.b  |-  ( ph  ->  B  e.  CC )
subeqxfrd.c  |-  ( ph  ->  C  e.  CC )
subeqxfrd.d  |-  ( ph  ->  D  e.  CC )
subeqxfrd.1  |-  ( ph  ->  ( A  -  B
)  =  ( C  -  D ) )
Assertion
Ref Expression
subeqxfrd  |-  ( ph  ->  ( A  -  C
)  =  ( B  -  D ) )

Proof of Theorem subeqxfrd
StepHypRef Expression
1 subeqxfrd.1 . . 3  |-  ( ph  ->  ( A  -  B
)  =  ( C  -  D ) )
21oveq1d 5934 . 2  |-  ( ph  ->  ( ( A  -  B )  +  ( B  -  C ) )  =  ( ( C  -  D )  +  ( B  -  C ) ) )
3 subeqxfrd.a . . 3  |-  ( ph  ->  A  e.  CC )
4 subeqxfrd.b . . 3  |-  ( ph  ->  B  e.  CC )
5 subeqxfrd.c . . 3  |-  ( ph  ->  C  e.  CC )
63, 4, 5npncand 8356 . 2  |-  ( ph  ->  ( ( A  -  B )  +  ( B  -  C ) )  =  ( A  -  C ) )
7 subeqxfrd.d . . 3  |-  ( ph  ->  D  e.  CC )
85, 7, 4npncan3d 8368 . 2  |-  ( ph  ->  ( ( C  -  D )  +  ( B  -  C ) )  =  ( B  -  D ) )
92, 6, 83eqtr3d 2234 1  |-  ( ph  ->  ( A  -  C
)  =  ( B  -  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2164  (class class class)co 5919   CCcc 7872    + caddc 7877    - cmin 8192
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-setind 4570  ax-resscn 7966  ax-1cn 7967  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-sub 8194
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator