Theorem addlsub 8301

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 5872	. . 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 8243	. . . 4 |- ( ph -> ( ( A + B ) - B ) = A )
5		eqtr2 2194	. . . . . 6 |- ( ( ( ( A + B ) - B ) = ( C - B ) /\ ( ( A + B ) - B ) = A ) -> ( C - B ) = A )
6	5	eqcomd 2181	. . . . 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 5872	. . 3 |- ( A = ( C - B ) -> ( A + B ) = ( ( C - B ) + B ) )
11		addlsub.c	. . . . 5 |- ( ph -> C e. CC )
12	11, 3	npcand 8246	. . . 4 |- ( ph -> ( ( C - B ) + B ) = C )
13		eqtr 2193	. . . . 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 1353   e. wcel 2146  (class class class)co 5865   CCcc 7784   + caddc 7789   - cmin 8102

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-setind 4530  ax-resscn 7878  ax-1cn 7879  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-distr 7890  ax-i2m1 7891  ax-0id 7894  ax-rnegex 7895  ax-cnre 7897

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-sub 8104

This theorem is referenced by:  addrsub  8302  subexsub  8303  nn0ob  11880  oddennn  12360