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Theorem bj-axemptylem 10399
Description: Lemma for bj-axempty 10400 and bj-axempty2 10401. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3911 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 10340 . . 3 BOUNDED
21bdsep1 10392 . 2 𝑥𝑦(𝑦𝑥 ↔ (𝑦𝑧 ∧ ⊥))
3 bi1 115 . . . 4 ((𝑦𝑥 ↔ (𝑦𝑧 ∧ ⊥)) → (𝑦𝑥 → (𝑦𝑧 ∧ ⊥)))
4 falimd 1274 . . . 4 ((𝑦𝑧 ∧ ⊥) → ⊥)
53, 4syl6 33 . . 3 ((𝑦𝑥 ↔ (𝑦𝑧 ∧ ⊥)) → (𝑦𝑥 → ⊥))
65alimi 1360 . 2 (∀𝑦(𝑦𝑥 ↔ (𝑦𝑧 ∧ ⊥)) → ∀𝑦(𝑦𝑥 → ⊥))
72, 6eximii 1509 1 𝑥𝑦(𝑦𝑥 → ⊥)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257  wfal 1264  wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443  ax-bd0 10320  ax-bdim 10321  ax-bdn 10324  ax-bdeq 10327  ax-bdsep 10391
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265
This theorem is referenced by:  bj-axempty  10400  bj-axempty2  10401
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