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Theorem bj-dtrucor2v 32436
Description: Version of dtrucor2 4867 with a dv condition, which does not require ax-13 2250 (nor ax-4 1734, ax-5 1841, ax-7 1937, ax-12 2049). (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 1892 . 2 𝑥 𝑥 = 𝑦
2 bj-dtrucor2v.1 . . . . 5 (𝑥 = 𝑦𝑥𝑦)
32necon2bi 2826 . . . 4 (𝑥 = 𝑦 → ¬ 𝑥 = 𝑦)
4 pm2.01 180 . . . 4 ((𝑥 = 𝑦 → ¬ 𝑥 = 𝑦) → ¬ 𝑥 = 𝑦)
53, 4ax-mp 5 . . 3 ¬ 𝑥 = 𝑦
65nex 1728 . 2 ¬ ∃𝑥 𝑥 = 𝑦
71, 6pm2.24ii 117 1 (𝜑 ∧ ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wex 1701  wne 2796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-6 1890
This theorem depends on definitions:  df-bi 197  df-ex 1702  df-ne 2797
This theorem is referenced by: (None)
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