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Theorem bj-axemptylem 13413
 Description: Lemma for bj-axempty 13414 and bj-axempty2 13415. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4086 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 13354 . . 3 BOUNDED
21bdsep1 13406 . 2 𝑥𝑦(𝑦𝑥 ↔ (𝑦𝑧 ∧ ⊥))
3 biimp 117 . . . 4 ((𝑦𝑥 ↔ (𝑦𝑧 ∧ ⊥)) → (𝑦𝑥 → (𝑦𝑧 ∧ ⊥)))
4 falimd 1347 . . . 4 ((𝑦𝑧 ∧ ⊥) → ⊥)
53, 4syl6 33 . . 3 ((𝑦𝑥 ↔ (𝑦𝑧 ∧ ⊥)) → (𝑦𝑥 → ⊥))
65alimi 1432 . 2 (∀𝑦(𝑦𝑥 ↔ (𝑦𝑧 ∧ ⊥)) → ∀𝑦(𝑦𝑥 → ⊥))
72, 6eximii 1579 1 𝑥𝑦(𝑦𝑥 → ⊥)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1330  ⊥wfal 1337  ∃wex 1469 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 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1487  ax-ial 1511  ax-bd0 13334  ax-bdim 13335  ax-bdn 13338  ax-bdeq 13341  ax-bdsep 13405 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338 This theorem is referenced by:  bj-axempty  13414  bj-axempty2  13415
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