Theorem subaddeqd 8395

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 5937	. . 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 8177	. . . 4 |- ( ph -> ( C + D ) = ( D + C ) )
6	5	oveq1d 5937	. . 3 |- ( ph -> ( ( C + D ) - ( D + B ) ) = ( ( D + C ) - ( D + B ) ) )
7	2, 6	eqtrd 2229	. 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 8375	. 2 |- ( ph -> ( ( A + B ) - ( D + B ) ) = ( A - D ) )
11	4, 3, 9	pnpcand 8374	. 2 |- ( ph -> ( ( D + C ) - ( D + B ) ) = ( C - B ) )
12	7, 10, 11	3eqtr3d 2237	1 |- ( ph -> ( A - D ) = ( C - B ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167  (class class class)co 5922   CCcc 7877   + caddc 7882   - cmin 8197

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-setind 4573  ax-resscn 7971  ax-1cn 7972  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-sub 8199

This theorem is referenced by: (None)