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Theorem bj-nnfalt 34927
Description: See nfal 2320 and bj-nfalt 34872. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nnfalt (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦𝑥𝜑)

Proof of Theorem bj-nnfalt
StepHypRef Expression
1 df-bj-nnf 34885 . . . 4 (Ⅎ'𝑦𝜑 ↔ ((∃𝑦𝜑𝜑) ∧ (𝜑 → ∀𝑦𝜑)))
21albii 1825 . . 3 (∀𝑥Ⅎ'𝑦𝜑 ↔ ∀𝑥((∃𝑦𝜑𝜑) ∧ (𝜑 → ∀𝑦𝜑)))
3 simpl 482 . . . . 5 (((∃𝑦𝜑𝜑) ∧ (𝜑 → ∀𝑦𝜑)) → (∃𝑦𝜑𝜑))
43alimi 1817 . . . 4 (∀𝑥((∃𝑦𝜑𝜑) ∧ (𝜑 → ∀𝑦𝜑)) → ∀𝑥(∃𝑦𝜑𝜑))
5 bj-nnflemea 34926 . . . 4 (∀𝑥(∃𝑦𝜑𝜑) → (∃𝑦𝑥𝜑 → ∀𝑥𝜑))
64, 5syl 17 . . 3 (∀𝑥((∃𝑦𝜑𝜑) ∧ (𝜑 → ∀𝑦𝜑)) → (∃𝑦𝑥𝜑 → ∀𝑥𝜑))
72, 6sylbi 216 . 2 (∀𝑥Ⅎ'𝑦𝜑 → (∃𝑦𝑥𝜑 → ∀𝑥𝜑))
8 simpr 484 . . . . 5 (((∃𝑦𝜑𝜑) ∧ (𝜑 → ∀𝑦𝜑)) → (𝜑 → ∀𝑦𝜑))
98alimi 1817 . . . 4 (∀𝑥((∃𝑦𝜑𝜑) ∧ (𝜑 → ∀𝑦𝜑)) → ∀𝑥(𝜑 → ∀𝑦𝜑))
10 bj-nnflemaa 34923 . . . 4 (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝑥𝜑))
119, 10syl 17 . . 3 (∀𝑥((∃𝑦𝜑𝜑) ∧ (𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝑥𝜑))
122, 11sylbi 216 . 2 (∀𝑥Ⅎ'𝑦𝜑 → (∀𝑥𝜑 → ∀𝑦𝑥𝜑))
13 df-bj-nnf 34885 . 2 (Ⅎ'𝑦𝑥𝜑 ↔ ((∃𝑦𝑥𝜑 → ∀𝑥𝜑) ∧ (∀𝑥𝜑 → ∀𝑦𝑥𝜑)))
147, 12, 13sylanbrc 582 1 (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1539  wex 1785  Ⅎ'wnnf 34884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-10 2140  ax-11 2157  ax-12 2174
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1786  df-bj-nnf 34885
This theorem is referenced by:  bj-pm11.53v  34938
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