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Theorem dtrucor2 4931
Description: The theorem form of the deduction dtrucor 4930 leads to a contradiction, as mentioned in the "Wrong!" example at mmdeduction.html#bad. (Contributed by NM, 20-Oct-2007.)
Hypothesis
Ref Expression
dtrucor2.1 (𝑥 = 𝑦𝑥𝑦)
Assertion
Ref Expression
dtrucor2 (𝜑 ∧ ¬ 𝜑)

Proof of Theorem dtrucor2
StepHypRef Expression
1 ax6e 2286 . 2 𝑥 𝑥 = 𝑦
2 dtrucor2.1 . . . . 5 (𝑥 = 𝑦𝑥𝑦)
32necon2bi 2853 . . . 4 (𝑥 = 𝑦 → ¬ 𝑥 = 𝑦)
4 pm2.01 180 . . . 4 ((𝑥 = 𝑦 → ¬ 𝑥 = 𝑦) → ¬ 𝑥 = 𝑦)
53, 4ax-mp 5 . . 3 ¬ 𝑥 = 𝑦
65nex 1771 . 2 ¬ ∃𝑥 𝑥 = 𝑦
71, 6pm2.24ii 117 1 (𝜑 ∧ ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wex 1744  wne 2823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-ne 2824
This theorem is referenced by: (None)
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