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Theorem dfoddfin2 4513
 Description: Alternate definition of odd number. (Contributed by SF, 25-Jan-2015.)
Assertion
Ref Expression
dfoddfin2 Oddfin = {x n Nn (x = ((n +c n) +c 1c) ((n +c n) +c 1c) ≠ )}
Distinct variable group:   x,n

Proof of Theorem dfoddfin2
StepHypRef Expression
1 df-oddfin 4445 . 2 Oddfin = {x (n Nn x = ((n +c n) +c 1c) x)}
2 r19.41v 2764 . . . 4 (n Nn (x = ((n +c n) +c 1c) x) ↔ (n Nn x = ((n +c n) +c 1c) x))
3 neeq1 2524 . . . . . 6 (x = ((n +c n) +c 1c) → (x ↔ ((n +c n) +c 1c) ≠ ))
43pm5.32i 618 . . . . 5 ((x = ((n +c n) +c 1c) x) ↔ (x = ((n +c n) +c 1c) ((n +c n) +c 1c) ≠ ))
54rexbii 2639 . . . 4 (n Nn (x = ((n +c n) +c 1c) x) ↔ n Nn (x = ((n +c n) +c 1c) ((n +c n) +c 1c) ≠ ))
62, 5bitr3i 242 . . 3 ((n Nn x = ((n +c n) +c 1c) x) ↔ n Nn (x = ((n +c n) +c 1c) ((n +c n) +c 1c) ≠ ))
76abbii 2465 . 2 {x (n Nn x = ((n +c n) +c 1c) x)} = {x n Nn (x = ((n +c n) +c 1c) ((n +c n) +c 1c) ≠ )}
81, 7eqtri 2373 1 Oddfin = {x n Nn (x = ((n +c n) +c 1c) ((n +c n) +c 1c) ≠ )}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   = wceq 1642  {cab 2339   ≠ wne 2516  ∃wrex 2615  ∅c0 3550  1cc1c 4134   Nn cnnc 4373   +c cplc 4375   Oddfin coddfin 4437 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-ne 2518  df-rex 2620  df-oddfin 4445 This theorem is referenced by:  evenodddisj  4516
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