Theorem subeqxfrd 8350

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

Step	Hyp	Ref	Expression
1		subeqxfrd.1	. . 3 |- ( ph -> ( A - B ) = ( C - D ) )
2	1	oveq1d 5911	. 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 )
6	3, 4, 5	npncand 8322	. 2 |- ( ph -> ( ( A - B ) + ( B - C ) ) = ( A - C ) )
7		subeqxfrd.d	. . 3 |- ( ph -> D e. CC )
8	5, 7, 4	npncan3d 8334	. 2 |- ( ph -> ( ( C - D ) + ( B - C ) ) = ( B - D ) )
9	2, 6, 8	3eqtr3d 2230	1 |- ( ph -> ( A - C ) = ( B - D ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2160  (class class class)co 5896   CCcc 7839   + caddc 7844   - cmin 8158

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 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-setind 4554  ax-resscn 7933  ax-1cn 7934  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-addcom 7941  ax-addass 7943  ax-distr 7945  ax-i2m1 7946  ax-0id 7949  ax-rnegex 7950  ax-cnre 7952

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-sub 8160

This theorem is referenced by: (None)