Theorem bj-19.23t 34952

Description: Statement 19.23t 2203 proved from modalK (obsoleting 19.23v 1945). (Contributed by BJ, 2-Dec-2023.)

Assertion

Ref	Expression
bj-19.23t	⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))

Proof of Theorem bj-19.23t

Step	Hyp	Ref	Expression
1		bj-nnf-exlim 34938	. 2 ⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜓)))
2		bj-nnfa 34910	. . . 4 ⊢ (Ⅎ'𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
3	2	imim2d 57	. . 3 ⊢ (Ⅎ'𝑥𝜓 → ((∃𝑥𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
4		19.38 1841	. . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
5	3, 4	syl6 35	. 2 ⊢ (Ⅎ'𝑥𝜓 → ((∃𝑥𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)))
6	1, 5	impbid 211	1 ⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 205  ∀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

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-bj-nnf 34906

This theorem is referenced by:  bj-pm11.53vw  34958  bj-equsvt  34961