Theorem pnpncand 8418

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 8354	. . . 4 |- ( ph -> ( B - C ) e. CC )
5	1, 4	addcld 8063	. . 3 |- ( ph -> ( A + ( B - C ) ) e. CC )
6	5, 2, 3	subsub2d 8383	. 2 |- ( ph -> ( ( A + ( B - C ) ) - ( B - C ) ) = ( ( A + ( B - C ) ) + ( C - B ) ) )
7	1, 4	pncand 8355	. 2 |- ( ph -> ( ( A + ( B - C ) ) - ( B - C ) ) = A )
8	6, 7	eqtr3d 2231	1 |- ( ph -> ( ( A + ( B - C ) ) + ( C - B ) ) = A )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167  (class class class)co 5925   CCcc 7894   + caddc 7899   - cmin 8214

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 4152  ax-pow 4208  ax-pr 4243  ax-setind 4574  ax-resscn 7988  ax-1cn 7989  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007

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 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-sub 8216

This theorem is referenced by: (None)