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Theorem bj-dtrucor2v 32926
 Description: Version of dtrucor2 4931 with a dv condition, which does not require ax-13 2282 (nor ax-4 1777, ax-5 1879, ax-7 1981, ax-12 2087). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-dtrucor2v.1 (𝑥 = 𝑦𝑥𝑦)
Assertion
Ref Expression
bj-dtrucor2v (𝜑 ∧ ¬ 𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-dtrucor2v
StepHypRef Expression
1 ax6ev 1947 . 2 𝑥 𝑥 = 𝑦
2 bj-dtrucor2v.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-6 1945 This theorem depends on definitions:  df-bi 197  df-ex 1745  df-ne 2824 This theorem is referenced by: (None)
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