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Theorem necon3bid 2261
 Description: Deduction from equality to inequality. (Contributed by NM, 23-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypothesis
Ref Expression
necon3bid.1 (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))
Assertion
Ref Expression
necon3bid (𝜑 → (𝐴𝐵𝐶𝐷))

Proof of Theorem necon3bid
StepHypRef Expression
1 df-ne 2221 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 necon3bid.1 . . 3 (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))
32necon3bbid 2260 . 2 (𝜑 → (¬ 𝐴 = 𝐵𝐶𝐷))
41, 3syl5bb 185 1 (𝜑 → (𝐴𝐵𝐶𝐷))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 102   = 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:  nebidc  2300  addneintrd  7262  addneintr2d  7263  negne0bd  7378  negned  7382  subne0d  7394  subne0ad  7396  subneintrd  7429  subneintr2d  7431  qapne  8671  xrlttri3  8819  sqne0  9485  cjne0  9736  absne0d  10014
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