Theorem dtrucor2 5333

Description: The theorem form of the deduction dtrucor 5332 leads to a contradiction, as mentioned in the "Wrong!" example at mmdeduction.html#bad 5332. Usage of this theorem is discouraged because it depends on ax-13 2401. (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 2412	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		dtrucor2.1	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 ≠ 𝑦)
3	2	necon2bi 2985	. . . 4 ⊢ (𝑥 = 𝑦 → ¬ 𝑥 = 𝑦)
4		pm2.01 190	. . . 4 ⊢ ((𝑥 = 𝑦 → ¬ 𝑥 = 𝑦) → ¬ 𝑥 = 𝑦)
5	3, 4	ax-mp 5	. . 3 ⊢ ¬ 𝑥 = 𝑦
6	5	nex 1833	. 2 ⊢ ¬ ∃𝑥 𝑥 = 𝑦
7	1, 6	pm2.24ii 121	1 ⊢ (𝜑 ∧ ¬ 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∃wex 1812   ≠ wne 2955

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2213  ax-13 2401

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-ne 2956

This theorem is used by: (None)