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Theorem dtrucor2 5254
 Description: The theorem form of the deduction dtrucor 5253 leads to a contradiction, as mentioned in the "Wrong!" example at mmdeduction.html#bad 5253. Usage of this theorem is discouraged because it depends on ax-13 2392. (Contributed by NM, 20-Oct-2007.) (New usage is discouraged.)
Hypothesis
Ref Expression
dtrucor2.1 (𝑥 = 𝑦𝑥𝑦)
Assertion
Ref Expression
dtrucor2 (𝜑 ∧ ¬ 𝜑)

Proof of Theorem dtrucor2
StepHypRef Expression
1 ax6e 2403 . 2 𝑥 𝑥 = 𝑦
2 dtrucor2.1 . . . . 5 (𝑥 = 𝑦𝑥𝑦)
32necon2bi 3043 . . . 4 (𝑥 = 𝑦 → ¬ 𝑥 = 𝑦)
4 pm2.01 192 . . . 4 ((𝑥 = 𝑦 → ¬ 𝑥 = 𝑦) → ¬ 𝑥 = 𝑦)
53, 4ax-mp 5 . . 3 ¬ 𝑥 = 𝑦
65nex 1802 . 2 ¬ ∃𝑥 𝑥 = 𝑦
71, 6pm2.24ii 120 1 (𝜑 ∧ ¬ 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∃wex 1781   ≠ wne 3013 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-12 2179  ax-13 2392 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-ne 3014 This theorem is referenced by: (None)
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