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Theorem bj-hbext 31888
 Description: Closed form of hbex 2142. (Contributed by BJ, 10-Oct-2019.)
Assertion
Ref Expression
bj-hbext (∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑦𝜑 → ∀𝑥𝑦𝜑))

Proof of Theorem bj-hbext
StepHypRef Expression
1 nfa2 2027 . . . 4 𝑥𝑦𝑥(𝜑 → ∀𝑥𝜑)
2 hbnt 2129 . . . . . 6 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
32alimi 1730 . . . . 5 (∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑦𝜑 → ∀𝑥 ¬ 𝜑))
4 bj-hbalt 31858 . . . . 5 (∀𝑦𝜑 → ∀𝑥 ¬ 𝜑) → (∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑))
53, 4syl 17 . . . 4 (∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑))
61, 5alrimi 2069 . . 3 (∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑))
7 hbnt 2129 . . 3 (∀𝑥(∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑) → (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑))
86, 7syl 17 . 2 (∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑))
9 df-ex 1696 . . 3 (∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑)
109bicomi 213 . 2 (¬ ∀𝑦 ¬ 𝜑 ↔ ∃𝑦𝜑)
1110albii 1737 . 2 (∀𝑥 ¬ ∀𝑦 ¬ 𝜑 ↔ ∀𝑥𝑦𝜑)
128, 10, 113imtr3g 283 1 (∀𝑦𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑦𝜑 → ∀𝑥𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701 This theorem is referenced by:  bj-nfext  31890
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