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Theorem neanior 2427
Description: A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
Assertion
Ref Expression
neanior ((𝐴𝐵𝐶𝐷) ↔ ¬ (𝐴 = 𝐵𝐶 = 𝐷))

Proof of Theorem neanior
StepHypRef Expression
1 df-ne 2341 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 df-ne 2341 . . 3 (𝐶𝐷 ↔ ¬ 𝐶 = 𝐷)
31, 2anbi12i 457 . 2 ((𝐴𝐵𝐶𝐷) ↔ (¬ 𝐴 = 𝐵 ∧ ¬ 𝐶 = 𝐷))
4 pm4.56 775 . 2 ((¬ 𝐴 = 𝐵 ∧ ¬ 𝐶 = 𝐷) ↔ ¬ (𝐴 = 𝐵𝐶 = 𝐷))
53, 4bitri 183 1 ((𝐴𝐵𝐶𝐷) ↔ ¬ (𝐴 = 𝐵𝐶 = 𝐷))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wb 104  wo 703   = wceq 1348  wne 2340
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
This theorem depends on definitions:  df-bi 116  df-ne 2341
This theorem is referenced by:  nelpri  3605  nelprd  3607  eldifpr  3608  0nelop  4231  lcmgcd  12021  lcmdvds  12022  lgsdirnn0  13703  lgsdinn0  13704
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