Theorem bj-19.21t 34688

Description: Statement 19.21t 2204 proved from modalK (obsoleting 19.21v 1947). (Contributed by BJ, 2-Dec-2023.)

Assertion

Ref	Expression
bj-19.21t	⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Proof of Theorem bj-19.21t

Step	Hyp	Ref	Expression
1		bj-nnf-alrim 34674	. 2 ⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
2		bj-nnfe 34650	. . . 4 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑))
3	2	imim1d 82	. . 3 ⊢ (Ⅎ'𝑥𝜑 → ((𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
4		19.38 1846	. . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
5	3, 4	syl6 35	. 2 ⊢ (Ⅎ'𝑥𝜑 → ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓)))
6	1, 5	impbid 215	1 ⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1541  ∃wex 1787  Ⅎ'wnnf 34642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-bj-nnf 34643

This theorem is referenced by:  bj-pm11.53vw  34695