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Theorem bj-axemptylem 10012
Description: Lemma for bj-axempty 10013 and bj-axempty2 10014. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3883 instead. (New usage is discouraged.)
Assertion
Ref Expression
bj-axemptylem 𝑥𝑦(𝑦𝑥 → ⊥)
Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-axemptylem
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bdfal 9953 . . 3 BOUNDED
21bdsep1 10005 . 2 𝑥𝑦(𝑦𝑥 ↔ (𝑦𝑧 ∧ ⊥))
3 bi1 111 . . . 4 ((𝑦𝑥 ↔ (𝑦𝑧 ∧ ⊥)) → (𝑦𝑥 → (𝑦𝑧 ∧ ⊥)))
4 falimd 1258 . . . 4 ((𝑦𝑧 ∧ ⊥) → ⊥)
53, 4syl6 29 . . 3 ((𝑦𝑥 ↔ (𝑦𝑧 ∧ ⊥)) → (𝑦𝑥 → ⊥))
65alimi 1344 . 2 (∀𝑦(𝑦𝑥 ↔ (𝑦𝑧 ∧ ⊥)) → ∀𝑦(𝑦𝑥 → ⊥))
72, 6eximii 1493 1 𝑥𝑦(𝑦𝑥 → ⊥)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  wal 1241  wfal 1248  wex 1381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-bd0 9933  ax-bdim 9934  ax-bdn 9937  ax-bdeq 9940  ax-bdsep 10004
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249
This theorem is referenced by:  bj-axempty  10013  bj-axempty2  10014
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