Theorem pnpncand 8708

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 8644	. . . 4 |- ( ph -> ( B - C ) e. CC )
5	1, 4	addcld 8351	. . 3 |- ( ph -> ( A + ( B - C ) ) e. CC )
6	5, 2, 3	subsub2d 8673	. 2 |- ( ph -> ( ( A + ( B - C ) ) - ( B - C ) ) = ( ( A + ( B - C ) ) + ( C - B ) ) )
7	1, 4	pncand 8645	. 2 |- ( ph -> ( ( A + ( B - C ) ) - ( B - C ) ) = A )
8	6, 7	eqtr3d 2273	1 |- ( ph -> ( ( A + ( B - C ) ) + ( C - B ) ) = A )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209  (class class class)co 6088   CCcc 8183   + caddc 8188   - cmin 8504

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 4251  ax-pow 4313  ax-pr 4348  ax-setind 4686  ax-resscn 8277  ax-1cn 8278  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296

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 4132  df-opab 4194  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-iota 5340  df-fun 5382  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-sub 8506

This theorem is used by: (None)