Theorem subeqxfrd 8696

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 6103	. 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 8668	. 2 |- ( ph -> ( ( A - B ) + ( B - C ) ) = ( A - C ) )
7		subeqxfrd.d	. . 3 |- ( ph -> D e. CC )
8	5, 7, 4	npncan3d 8680	. 2 |- ( ph -> ( ( C - D ) + ( B - C ) ) = ( B - D ) )
9	2, 6, 8	3eqtr3d 2279	1 |- ( ph -> ( A - C ) = ( B - D ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209  (class class class)co 6088   CCcc 8183   + caddc 8188   - cmin 8504

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-setind 4686  ax-resscn 8277  ax-1cn 8278  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-iota 5340  df-fun 5382  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-sub 8506

This theorem is used by: (None)