Theorem addlsub 8268

Description: Left-subtraction: Subtraction of the left summand from the result of an addition. (Contributed by BJ, 6-Jun-2019.)

Hypotheses

Ref	Expression
addlsub.a	|- ( ph -> A e. CC )
addlsub.b	|- ( ph -> B e. CC )
addlsub.c	|- ( ph -> C e. CC )

Assertion

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

Proof of Theorem addlsub

Step	Hyp	Ref	Expression
1		oveq1 5849	. . 3 |- ( ( A + B ) = C -> ( ( A + B ) - B ) = ( C - B ) )
2		addlsub.a	. . . . 5 |- ( ph -> A e. CC )
3		addlsub.b	. . . . 5 |- ( ph -> B e. CC )
4	2, 3	pncand 8210	. . . 4 |- ( ph -> ( ( A + B ) - B ) = A )
5		eqtr2 2184	. . . . . 6 |- ( ( ( ( A + B ) - B ) = ( C - B ) /\ ( ( A + B ) - B ) = A ) -> ( C - B ) = A )
6	5	eqcomd 2171	. . . . 5 |- ( ( ( ( A + B ) - B ) = ( C - B ) /\ ( ( A + B ) - B ) = A ) -> A = ( C - B ) )
7	6	a1i 9	. . . 4 |- ( ph -> ( ( ( ( A + B ) - B ) = ( C - B ) /\ ( ( A + B ) - B ) = A ) -> A = ( C - B ) ) )
8	4, 7	mpan2d 425	. . 3 |- ( ph -> ( ( ( A + B ) - B ) = ( C - B ) -> A = ( C - B ) ) )
9	1, 8	syl5 32	. 2 |- ( ph -> ( ( A + B ) = C -> A = ( C - B ) ) )
10		oveq1 5849	. . 3 |- ( A = ( C - B ) -> ( A + B ) = ( ( C - B ) + B ) )
11		addlsub.c	. . . . 5 |- ( ph -> C e. CC )
12	11, 3	npcand 8213	. . . 4 |- ( ph -> ( ( C - B ) + B ) = C )
13		eqtr 2183	. . . . 5 |- ( ( ( A + B ) = ( ( C - B ) + B ) /\ ( ( C - B ) + B ) = C ) -> ( A + B ) = C )
14	13	a1i 9	. . . 4 |- ( ph -> ( ( ( A + B ) = ( ( C - B ) + B ) /\ ( ( C - B ) + B ) = C ) -> ( A + B ) = C ) )
15	12, 14	mpan2d 425	. . 3 |- ( ph -> ( ( A + B ) = ( ( C - B ) + B ) -> ( A + B ) = C ) )
16	10, 15	syl5 32	. 2 |- ( ph -> ( A = ( C - B ) -> ( A + B ) = C ) )
17	9, 16	impbid 128	1 |- ( ph -> ( ( A + B ) = C <-> A = ( C - B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136  (class class class)co 5842   CCcc 7751   + caddc 7756   - cmin 8069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514  ax-resscn 7845  ax-1cn 7846  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-sub 8071

This theorem is referenced by:  addrsub  8269  subexsub  8270  nn0ob  11845  oddennn  12325