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Theorem bj-axempty2 11785
Description: Axiom of the empty set from bounded separation, alternate version to bj-axempty 11784. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3965 instead. (New usage is discouraged.)
Assertion
Ref Expression
bj-axempty2 𝑥𝑦 ¬ 𝑦𝑥
Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-axempty2
StepHypRef Expression
1 bj-axemptylem 11783 . 2 𝑥𝑦(𝑦𝑥 → ⊥)
2 dfnot 1307 . . . 4 𝑦𝑥 ↔ (𝑦𝑥 → ⊥))
32albii 1404 . . 3 (∀𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥 → ⊥))
43exbii 1541 . 2 (∃𝑥𝑦 ¬ 𝑦𝑥 ↔ ∃𝑥𝑦(𝑦𝑥 → ⊥))
51, 4mpbir 144 1 𝑥𝑦 ¬ 𝑦𝑥
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1287  wfal 1294  wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472  ax-bd0 11704  ax-bdim 11705  ax-bdn 11708  ax-bdeq 11711  ax-bdsep 11775
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295
This theorem is referenced by: (None)
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