Theorem subaddeqd 7901

Description: Transfer two terms of a subtraction to an addition in an equality. (Contributed by Thierry Arnoux, 2-Feb-2020.)

Hypotheses

Ref	Expression
subaddeqd.a	|- ( ph -> A e. CC )
subaddeqd.b	|- ( ph -> B e. CC )
subaddeqd.c	|- ( ph -> C e. CC )
subaddeqd.d	|- ( ph -> D e. CC )
subaddeqd.1	|- ( ph -> ( A + B ) = ( C + D ) )

Assertion

Ref	Expression
subaddeqd	|- ( ph -> ( A - D ) = ( C - B ) )

Proof of Theorem subaddeqd

Step	Hyp	Ref	Expression
1		subaddeqd.1	. . . 4 |- ( ph -> ( A + B ) = ( C + D ) )
2	1	oveq1d 5681	. . 3 |- ( ph -> ( ( A + B ) - ( D + B ) ) = ( ( C + D ) - ( D + B ) ) )
3		subaddeqd.c	. . . . 5 |- ( ph -> C e. CC )
4		subaddeqd.d	. . . . 5 |- ( ph -> D e. CC )
5	3, 4	addcomd 7687	. . . 4 |- ( ph -> ( C + D ) = ( D + C ) )
6	5	oveq1d 5681	. . 3 |- ( ph -> ( ( C + D ) - ( D + B ) ) = ( ( D + C ) - ( D + B ) ) )
7	2, 6	eqtrd 2121	. 2 |- ( ph -> ( ( A + B ) - ( D + B ) ) = ( ( D + C ) - ( D + B ) ) )
8		subaddeqd.a	. . 3 |- ( ph -> A e. CC )
9		subaddeqd.b	. . 3 |- ( ph -> B e. CC )
10	8, 4, 9	pnpcan2d 7885	. 2 |- ( ph -> ( ( A + B ) - ( D + B ) ) = ( A - D ) )
11	4, 3, 9	pnpcand 7884	. 2 |- ( ph -> ( ( D + C ) - ( D + B ) ) = ( C - B ) )
12	7, 10, 11	3eqtr3d 2129	1 |- ( ph -> ( A - D ) = ( C - B ) )

Syntax hints:   -> wi 4   = wceq 1290   e. wcel 1439  (class class class)co 5666   CCcc 7402   + caddc 7407   - cmin 7707

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-setind 4366  ax-resscn 7491  ax-1cn 7492  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-addcom 7499  ax-addass 7501  ax-distr 7503  ax-i2m1 7504  ax-0id 7507  ax-rnegex 7508  ax-cnre 7510

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-iota 4993  df-fun 5030  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-sub 7709

This theorem is referenced by: (None)