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Theorem bj-19.21t 34232
 Description: Statement 19.21t 2204 proved from modalK (obsoleting 19.21v 1940). (Contributed by BJ, 2-Dec-2023.)
Assertion
Ref Expression
bj-19.21t (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Proof of Theorem bj-19.21t
StepHypRef Expression
1 bj-nnf-alrim 34218 . 2 (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))
2 bj-nnfe 34196 . . . 4 (Ⅎ'𝑥𝜑 → (∃𝑥𝜑𝜑))
32imim1d 82 . . 3 (Ⅎ'𝑥𝜑 → ((𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
4 19.38 1840 . . 3 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
53, 4syl6 35 . 2 (Ⅎ'𝑥𝜑 → ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓)))
61, 5impbid 215 1 (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  ∃wex 1781  Ⅎ'wnnf 34189 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-bj-nnf 34190 This theorem is referenced by: (None)
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