Theorem subeqxfrd 8541

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 6032	. 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 8513	. 2 |- ( ph -> ( ( A - B ) + ( B - C ) ) = ( A - C ) )
7		subeqxfrd.d	. . 3 |- ( ph -> D e. CC )
8	5, 7, 4	npncan3d 8525	. 2 |- ( ph -> ( ( C - D ) + ( B - C ) ) = ( B - D ) )
9	2, 6, 8	3eqtr3d 2272	1 |- ( ph -> ( A - C ) = ( B - D ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202  (class class class)co 6017   CCcc 8029   + caddc 8034   - cmin 8349

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-setind 4635  ax-resscn 8123  ax-1cn 8124  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-sub 8351

This theorem is referenced by: (None)