Theorem subaddeqd 8695

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 6100	. . 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 8477	. . . 4 |- ( ph -> ( C + D ) = ( D + C ) )
6	5	oveq1d 6100	. . 3 |- ( ph -> ( ( C + D ) - ( D + B ) ) = ( ( D + C ) - ( D + B ) ) )
7	2, 6	eqtrd 2271	. 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 8675	. 2 |- ( ph -> ( ( A + B ) - ( D + B ) ) = ( A - D ) )
11	4, 3, 9	pnpcand 8674	. 2 |- ( ph -> ( ( D + C ) - ( D + B ) ) = ( C - B ) )
12	7, 10, 11	3eqtr3d 2279	1 |- ( ph -> ( A - D ) = ( C - B ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209  (class class class)co 6085   CCcc 8177   + caddc 8182   - cmin 8497

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 4249  ax-pow 4311  ax-pr 4346  ax-setind 4684  ax-resscn 8271  ax-1cn 8272  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

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 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-sub 8499

This theorem is used by: (None)