Theorem dtrucor2 5303

Description: The theorem form of the deduction dtrucor 5302 leads to a contradiction, as mentioned in the "Wrong!" example at mmdeduction.html#bad 5302. Usage of this theorem is discouraged because it depends on ax-13 2382. (Contributed by NM, 20-Oct-2007.) (New usage is discouraged.)

Hypothesis

Ref	Expression
dtrucor2.1	⊢ (𝑥 = 𝑦 → 𝑥 ≠ 𝑦)

Assertion

Ref	Expression
dtrucor2	⊢ (𝜑 ∧ ¬ 𝜑)

Proof of Theorem dtrucor2

Step	Hyp	Ref	Expression
1		ax6e 2393	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		dtrucor2.1	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 ≠ 𝑦)
3	2	necon2bi 2966	. . . 4 ⊢ (𝑥 = 𝑦 → ¬ 𝑥 = 𝑦)
4		pm2.01 189	. . . 4 ⊢ ((𝑥 = 𝑦 → ¬ 𝑥 = 𝑦) → ¬ 𝑥 = 𝑦)
5	3, 4	ax-mp 5	. . 3 ⊢ ¬ 𝑥 = 𝑦
6	5	nex 1808	. 2 ⊢ ¬ ∃𝑥 𝑥 = 𝑦
7	1, 6	pm2.24ii 120	1 ⊢ (𝜑 ∧ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397  ∃wex 1787   ≠ wne 2936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-12 2191  ax-13 2382

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-ne 2937

This theorem is referenced by: (None)