Theorem addlsub 8516

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 6008	. . 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 8458	. . . 4 |- ( ph -> ( ( A + B ) - B ) = A )
5		eqtr2 2248	. . . . . 6 |- ( ( ( ( A + B ) - B ) = ( C - B ) /\ ( ( A + B ) - B ) = A ) -> ( C - B ) = A )
6	5	eqcomd 2235	. . . . 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 428	. . 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 6008	. . 3 |- ( A = ( C - B ) -> ( A + B ) = ( ( C - B ) + B ) )
11		addlsub.c	. . . . 5 |- ( ph -> C e. CC )
12	11, 3	npcand 8461	. . . 4 |- ( ph -> ( ( C - B ) + B ) = C )
13		eqtr 2247	. . . . 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 428	. . 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 129	1 |- ( ph -> ( ( A + B ) = C <-> A = ( C - B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200  (class class class)co 6001   CCcc 7997   + caddc 8002   - cmin 8317

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-setind 4629  ax-resscn 8091  ax-1cn 8092  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-sub 8319

This theorem is referenced by:  addrsub  8517  subexsub  8518  nn0ob  12419  oddennn  12963