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Theorem nonconne 2232
Description: Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)
Assertion
Ref Expression
nonconne ¬ (𝐴 = 𝐵𝐴𝐵)

Proof of Theorem nonconne
StepHypRef Expression
1 pm3.24 637 . 2 ¬ (𝐴 = 𝐵 ∧ ¬ 𝐴 = 𝐵)
2 df-ne 2221 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
32anbi2i 438 . 2 ((𝐴 = 𝐵𝐴𝐵) ↔ (𝐴 = 𝐵 ∧ ¬ 𝐴 = 𝐵))
41, 3mtbir 606 1 ¬ (𝐴 = 𝐵𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 101   = wceq 1259  wne 2220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555
This theorem depends on definitions:  df-bi 114  df-ne 2221
This theorem is referenced by: (None)
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