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Theorem subadd 6991
Description: Relationship between subtraction and addition. (Contributed by NM, 20-Jan-1997.) (Revised by Mario Carneiro, 21-Dec-2013.)
Assertion
Ref Expression
subadd ((A B 𝐶 ℂ) → ((AB) = 𝐶 ↔ (B + 𝐶) = A))

Proof of Theorem subadd
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 subval 6980 . . . 4 ((A B ℂ) → (AB) = (x ℂ (B + x) = A))
21eqeq1d 2045 . . 3 ((A B ℂ) → ((AB) = 𝐶 ↔ (x ℂ (B + x) = A) = 𝐶))
323adant3 923 . 2 ((A B 𝐶 ℂ) → ((AB) = 𝐶 ↔ (x ℂ (B + x) = A) = 𝐶))
4 negeu 6979 . . . . 5 ((B A ℂ) → ∃!x ℂ (B + x) = A)
5 oveq2 5463 . . . . . . 7 (x = 𝐶 → (B + x) = (B + 𝐶))
65eqeq1d 2045 . . . . . 6 (x = 𝐶 → ((B + x) = A ↔ (B + 𝐶) = A))
76riota2 5433 . . . . 5 ((𝐶 ∃!x ℂ (B + x) = A) → ((B + 𝐶) = A ↔ (x ℂ (B + x) = A) = 𝐶))
84, 7sylan2 270 . . . 4 ((𝐶 (B A ℂ)) → ((B + 𝐶) = A ↔ (x ℂ (B + x) = A) = 𝐶))
983impb 1099 . . 3 ((𝐶 B A ℂ) → ((B + 𝐶) = A ↔ (x ℂ (B + x) = A) = 𝐶))
1093com13 1108 . 2 ((A B 𝐶 ℂ) → ((B + 𝐶) = A ↔ (x ℂ (B + x) = A) = 𝐶))
113, 10bitr4d 180 1 ((A B 𝐶 ℂ) → ((AB) = 𝐶 ↔ (B + 𝐶) = A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  ∃!wreu 2302  crio 5410  (class class class)co 5455  cc 6689   + caddc 6694  cmin 6959
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220  ax-resscn 6755  ax-1cn 6756  ax-icn 6758  ax-addcl 6759  ax-addrcl 6760  ax-mulcl 6761  ax-addcom 6763  ax-addass 6765  ax-distr 6767  ax-i2m1 6768  ax-0id 6771  ax-rnegex 6772  ax-cnre 6774
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-sub 6961
This theorem is referenced by:  subadd2  6992  subsub23  6993  pncan  6994  pncan3  6996  addsubeq4  7003  subsub2  7015  renegcl  7048  subaddi  7074  subaddd  7116  fzen  8657  nn0ennn  8870
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