Theorem bj-19.23t 37739

Description: Statement 19.23t 2246 proved from modalK (obsoleting 19.23v 1974). (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 37737	. 2 ⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜓)))
2		bj-nnfa 37705	. . . 4 ⊢ (Ⅎ'𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
3	2	imim2d 58	. . 3 ⊢ (Ⅎ'𝑥𝜓 → ((∃𝑥𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
4		19.38 1871	. . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
5	3, 4	syl6 36	. 2 ⊢ (Ⅎ'𝑥𝜓 → ((∃𝑥𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)))
6	1, 5	impbid 215	1 ⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567  ∃wex 1811  Ⅎ'wnnf 37703

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-bj-nnf 37704

This theorem is used by:  bj-pm11.53vw  37744  bj-equsvt  37748