Theorem bj-dtrucor2v 36217

Description: Version of dtrucor2 5366 with a disjoint variable condition, which does not require ax-13 2366 (nor ax-4 1804, ax-5 1906, ax-7 2004, ax-12 2164). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-dtrucor2v.1	⊢ (𝑥 = 𝑦 → 𝑥 ≠ 𝑦)

Assertion

Ref	Expression
bj-dtrucor2v	⊢ (𝜑 ∧ ¬ 𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-dtrucor2v

Step	Hyp	Ref	Expression
1		ax6ev 1966	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		bj-dtrucor2v.1	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 ≠ 𝑦)
3	2	necon2bi 2966	. . . 4 ⊢ (𝑥 = 𝑦 → ¬ 𝑥 = 𝑦)
4		pm2.01 188	. . . 4 ⊢ ((𝑥 = 𝑦 → ¬ 𝑥 = 𝑦) → ¬ 𝑥 = 𝑦)
5	3, 4	ax-mp 5	. . 3 ⊢ ¬ 𝑥 = 𝑦
6	5	nex 1795	. 2 ⊢ ¬ ∃𝑥 𝑥 = 𝑦
7	1, 6	pm2.24ii 120	1 ⊢ (𝜑 ∧ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∃wex 1774   ≠ wne 2935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-6 1964

This theorem depends on definitions:  df-bi 206  df-ex 1775  df-ne 2936

This theorem is referenced by: (None)