Theorem pnpncand 8394

Description: Addition/subtraction cancellation law. (Contributed by Scott Fenton, 14-Dec-2017.)

Hypotheses

Ref	Expression
pnpncand.1	|- ( ph -> A e. CC )
pnpncand.2	|- ( ph -> B e. CC )
pnpncand.3	|- ( ph -> C e. CC )

Assertion

Ref	Expression
pnpncand	|- ( ph -> ( ( A + ( B - C ) ) + ( C - B ) ) = A )

Proof of Theorem pnpncand

Step	Hyp	Ref	Expression
1		pnpncand.1	. . . 4 |- ( ph -> A e. CC )
2		pnpncand.2	. . . . 5 |- ( ph -> B e. CC )
3		pnpncand.3	. . . . 5 |- ( ph -> C e. CC )
4	2, 3	subcld 8330	. . . 4 |- ( ph -> ( B - C ) e. CC )
5	1, 4	addcld 8039	. . 3 |- ( ph -> ( A + ( B - C ) ) e. CC )
6	5, 2, 3	subsub2d 8359	. 2 |- ( ph -> ( ( A + ( B - C ) ) - ( B - C ) ) = ( ( A + ( B - C ) ) + ( C - B ) ) )
7	1, 4	pncand 8331	. 2 |- ( ph -> ( ( A + ( B - C ) ) - ( B - C ) ) = A )
8	6, 7	eqtr3d 2228	1 |- ( ph -> ( ( A + ( B - C ) ) + ( C - B ) ) = A )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164  (class class class)co 5918   CCcc 7870   + caddc 7875   - cmin 8190

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-setind 4569  ax-resscn 7964  ax-1cn 7965  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-sub 8192

This theorem is referenced by: (None)