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Theorem axprlem2 5320
 Description: Lemma for axpr 5324. There exists a set to which all sets whose only members are empty sets belong. (Contributed by Rohan Ridenour, 9-Aug-2023.) (Revised by BJ, 13-Aug-2023.)
Assertion
Ref Expression
axprlem2 𝑥𝑦(∀𝑧𝑦𝑤 ¬ 𝑤𝑧𝑦𝑥)
Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Proof of Theorem axprlem2
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 ax-pow 5262 . . 3 𝑥𝑦(∀𝑧(𝑧𝑦𝑧𝑣) → 𝑦𝑥)
2 df-ral 3147 . . . . . . 7 (∀𝑧𝑦𝑤 ¬ 𝑤𝑧 ↔ ∀𝑧(𝑧𝑦 → ∀𝑤 ¬ 𝑤𝑧))
3 imim2 58 . . . . . . . 8 ((∀𝑤 ¬ 𝑤𝑧𝑧𝑣) → ((𝑧𝑦 → ∀𝑤 ¬ 𝑤𝑧) → (𝑧𝑦𝑧𝑣)))
43al2imi 1809 . . . . . . 7 (∀𝑧(∀𝑤 ¬ 𝑤𝑧𝑧𝑣) → (∀𝑧(𝑧𝑦 → ∀𝑤 ¬ 𝑤𝑧) → ∀𝑧(𝑧𝑦𝑧𝑣)))
52, 4syl5bi 243 . . . . . 6 (∀𝑧(∀𝑤 ¬ 𝑤𝑧𝑧𝑣) → (∀𝑧𝑦𝑤 ¬ 𝑤𝑧 → ∀𝑧(𝑧𝑦𝑧𝑣)))
65imim1d 82 . . . . 5 (∀𝑧(∀𝑤 ¬ 𝑤𝑧𝑧𝑣) → ((∀𝑧(𝑧𝑦𝑧𝑣) → 𝑦𝑥) → (∀𝑧𝑦𝑤 ¬ 𝑤𝑧𝑦𝑥)))
76alimdv 1910 . . . 4 (∀𝑧(∀𝑤 ¬ 𝑤𝑧𝑧𝑣) → (∀𝑦(∀𝑧(𝑧𝑦𝑧𝑣) → 𝑦𝑥) → ∀𝑦(∀𝑧𝑦𝑤 ¬ 𝑤𝑧𝑦𝑥)))
87eximdv 1911 . . 3 (∀𝑧(∀𝑤 ¬ 𝑤𝑧𝑧𝑣) → (∃𝑥𝑦(∀𝑧(𝑧𝑦𝑧𝑣) → 𝑦𝑥) → ∃𝑥𝑦(∀𝑧𝑦𝑤 ¬ 𝑤𝑧𝑦𝑥)))
91, 8mpi 20 . 2 (∀𝑧(∀𝑤 ¬ 𝑤𝑧𝑧𝑣) → ∃𝑥𝑦(∀𝑧𝑦𝑤 ¬ 𝑤𝑧𝑦𝑥))
10 axprlem1 5319 . 2 𝑣𝑧(∀𝑤 ¬ 𝑤𝑧𝑧𝑣)
119, 10exlimiiv 1925 1 𝑥𝑦(∀𝑧𝑦𝑤 ¬ 𝑤𝑧𝑦𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1528  ∃wex 1773  ∀wral 3142 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-nul 5206  ax-pow 5262 This theorem depends on definitions:  df-bi 208  df-ex 1774  df-ral 3147 This theorem is referenced by:  axpr  5324
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