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Theorem bj-axempty2 13929
Description: Axiom of the empty set from bounded separation, alternate version to bj-axempty 13928. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4115 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 13927 . 2 𝑥𝑦(𝑦𝑥 → ⊥)
2 dfnot 1366 . . . 4 𝑦𝑥 ↔ (𝑦𝑥 → ⊥))
32albii 1463 . . 3 (∀𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥 → ⊥))
43exbii 1598 . 2 (∃𝑥𝑦 ¬ 𝑦𝑥 ↔ ∃𝑥𝑦(𝑦𝑥 → ⊥))
51, 4mpbir 145 1 𝑥𝑦 ¬ 𝑦𝑥
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1346  wfal 1353  wex 1485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527  ax-bd0 13848  ax-bdim 13849  ax-bdn 13852  ax-bdeq 13855  ax-bdsep 13919
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354
This theorem is referenced by: (None)
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