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Theorem bj-nnfext 34949
Description: See nfex 2318 and bj-nfext 34894. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nnfext (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦𝑥𝜑)

Proof of Theorem bj-nnfext
StepHypRef Expression
1 df-bj-nnf 34906 . . . 4 (Ⅎ'𝑦𝜑 ↔ ((∃𝑦𝜑𝜑) ∧ (𝜑 → ∀𝑦𝜑)))
21albii 1822 . . 3 (∀𝑥Ⅎ'𝑦𝜑 ↔ ∀𝑥((∃𝑦𝜑𝜑) ∧ (𝜑 → ∀𝑦𝜑)))
3 simpl 483 . . . . 5 (((∃𝑦𝜑𝜑) ∧ (𝜑 → ∀𝑦𝜑)) → (∃𝑦𝜑𝜑))
43alimi 1814 . . . 4 (∀𝑥((∃𝑦𝜑𝜑) ∧ (𝜑 → ∀𝑦𝜑)) → ∀𝑥(∃𝑦𝜑𝜑))
5 bj-nnflemee 34945 . . . 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 485 . . . . 5 (((∃𝑦𝜑𝜑) ∧ (𝜑 → ∀𝑦𝜑)) → (𝜑 → ∀𝑦𝜑))
98alimi 1814 . . . 4 (∀𝑥((∃𝑦𝜑𝜑) ∧ (𝜑 → ∀𝑦𝜑)) → ∀𝑥(𝜑 → ∀𝑦𝜑))
10 bj-nnflemae 34946 . . . 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 34906 . 2 (Ⅎ'𝑦𝑥𝜑 ↔ ((∃𝑦𝑥𝜑 → ∃𝑥𝜑) ∧ (∃𝑥𝜑 → ∀𝑦𝑥𝜑)))
147, 12, 13sylanbrc 583 1 (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1537  wex 1782  Ⅎ'wnnf 34905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-bj-nnf 34906
This theorem is referenced by: (None)
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