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Theorem necon3bbid 2289
Description: Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)
Hypothesis
Ref Expression
necon3bbid.1 (𝜑 → (𝜓𝐴 = 𝐵))
Assertion
Ref Expression
necon3bbid (𝜑 → (¬ 𝜓𝐴𝐵))

Proof of Theorem necon3bbid
StepHypRef Expression
1 necon3bbid.1 . . . 4 (𝜑 → (𝜓𝐴 = 𝐵))
21bicomd 139 . . 3 (𝜑 → (𝐴 = 𝐵𝜓))
32necon3abid 2288 . 2 (𝜑 → (𝐴𝐵 ↔ ¬ 𝜓))
43bicomd 139 1 (𝜑 → (¬ 𝜓𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103   = wceq 1285  wne 2249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578
This theorem depends on definitions:  df-bi 115  df-ne 2250
This theorem is referenced by:  necon3bid  2290  eldifsn  3541  prmrp  10904
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