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Theorem bj-axempty2 12926
Description: Axiom of the empty set from bounded separation, alternate version to bj-axempty 12925. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4022 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 12924 . 2 𝑥𝑦(𝑦𝑥 → ⊥)
2 dfnot 1332 . . . 4 𝑦𝑥 ↔ (𝑦𝑥 → ⊥))
32albii 1429 . . 3 (∀𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥 → ⊥))
43exbii 1567 . 2 (∃𝑥𝑦 ¬ 𝑦𝑥 ↔ ∃𝑥𝑦(𝑦𝑥 → ⊥))
51, 4mpbir 145 1 𝑥𝑦 ¬ 𝑦𝑥
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1312  wfal 1319  wex 1451
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 586  ax-in2 587  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-ial 1497  ax-bd0 12845  ax-bdim 12846  ax-bdn 12849  ax-bdeq 12852  ax-bdsep 12916
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320
This theorem is referenced by: (None)
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