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Theorem 0cnALT 6978
Description: Alternate proof of 0cn 6797. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
0cnALT 0

Proof of Theorem 0cnALT
StepHypRef Expression
1 ax-icn 6758 . . 3 i
2 cnegex 6966 . . 3 (i ℂ → x ℂ (i + x) = 0)
31, 2ax-mp 7 . 2 x ℂ (i + x) = 0
4 addcl 6784 . . . . 5 ((i x ℂ) → (i + x) ℂ)
51, 4mpan 400 . . . 4 (x ℂ → (i + x) ℂ)
6 eleq1 2097 . . . 4 ((i + x) = 0 → ((i + x) ℂ ↔ 0 ℂ))
75, 6syl5ibcom 144 . . 3 (x ℂ → ((i + x) = 0 → 0 ℂ))
87rexlimiv 2421 . 2 (x ℂ (i + x) = 0 → 0 ℂ)
93, 8ax-mp 7 1 0
Colors of variables: wff set class
Syntax hints:   = wceq 1242   wcel 1390  wrex 2301  (class class class)co 5455  cc 6689  0cc0 6691  ici 6693   + caddc 6694
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-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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  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-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458
This theorem is referenced by: (None)
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