Theorem addlsub 8289

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 5860	. . 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 8231	. . . 4 |- ( ph -> ( ( A + B ) - B ) = A )
5		eqtr2 2189	. . . . . 6 |- ( ( ( ( A + B ) - B ) = ( C - B ) /\ ( ( A + B ) - B ) = A ) -> ( C - B ) = A )
6	5	eqcomd 2176	. . . . 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 426	. . 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 5860	. . 3 |- ( A = ( C - B ) -> ( A + B ) = ( ( C - B ) + B ) )
11		addlsub.c	. . . . 5 |- ( ph -> C e. CC )
12	11, 3	npcand 8234	. . . 4 |- ( ph -> ( ( C - B ) + B ) = C )
13		eqtr 2188	. . . . 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 426	. . 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 1348   e. wcel 2141  (class class class)co 5853   CCcc 7772   + caddc 7777   - cmin 8090

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521  ax-resscn 7866  ax-1cn 7867  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-sub 8092

This theorem is referenced by:  addrsub  8290  subexsub  8291  nn0ob  11867  oddennn  12347