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Theorem bj-axemptylem 12892
Description: Lemma for bj-axempty 12893 and bj-axempty2 12894. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4022 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 12833 . . 3 BOUNDED
21bdsep1 12885 . 2 𝑥𝑦(𝑦𝑥 ↔ (𝑦𝑧 ∧ ⊥))
3 bi1 117 . . . 4 ((𝑦𝑥 ↔ (𝑦𝑧 ∧ ⊥)) → (𝑦𝑥 → (𝑦𝑧 ∧ ⊥)))
4 falimd 1329 . . . 4 ((𝑦𝑧 ∧ ⊥) → ⊥)
53, 4syl6 33 . . 3 ((𝑦𝑥 ↔ (𝑦𝑧 ∧ ⊥)) → (𝑦𝑥 → ⊥))
65alimi 1414 . 2 (∀𝑦(𝑦𝑥 ↔ (𝑦𝑧 ∧ ⊥)) → ∀𝑦(𝑦𝑥 → ⊥))
72, 6eximii 1564 1 𝑥𝑦(𝑦𝑥 → ⊥)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  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 12813  ax-bdim 12814  ax-bdn 12817  ax-bdeq 12820  ax-bdsep 12884
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320
This theorem is referenced by:  bj-axempty  12893  bj-axempty2  12894
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