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Theorem subadd 6816
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 6805 . . . 4 ((A B ℂ) → (AB) = (x ℂ (B + x) = A))
21eqeq1d 2026 . . 3 ((A B ℂ) → ((AB) = 𝐶 ↔ (x ℂ (B + x) = A) = 𝐶))
323adant3 910 . 2 ((A B 𝐶 ℂ) → ((AB) = 𝐶 ↔ (x ℂ (B + x) = A) = 𝐶))
4 negeu 6804 . . . . 5 ((B A ℂ) → ∃!x ℂ (B + x) = A)
5 oveq2 5440 . . . . . . 7 (x = 𝐶 → (B + x) = (B + 𝐶))
65eqeq1d 2026 . . . . . 6 (x = 𝐶 → ((B + x) = A ↔ (B + 𝐶) = A))
76riota2 5410 . . . . 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 1084 . . 3 ((𝐶 B A ℂ) → ((B + 𝐶) = A ↔ (x ℂ (B + x) = A) = 𝐶))
1093com13 1093 . 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 871   = wceq 1226   wcel 1370  ∃!wreu 2282  crio 5388  (class class class)co 5432  cc 6522   + caddc 6527  cmin 6784
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-setind 4200  ax-resscn 6582  ax-1cn 6583  ax-icn 6585  ax-addcl 6586  ax-addrcl 6587  ax-mulcl 6588  ax-addcom 6590  ax-addass 6592  ax-distr 6594  ax-i2m1 6595  ax-0id 6598  ax-rnegex 6599  ax-cnre 6601
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-iota 4790  df-fun 4827  df-fv 4833  df-riota 5389  df-ov 5435  df-oprab 5436  df-mpt2 5437  df-sub 6786
This theorem is referenced by:  subadd2  6817  subsub23  6818  pncan  6819  pncan3  6821  addsubeq4  6828  subsub2  6840  renegcl  6873  subaddi  6899  subaddd  6941
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