Theorem bj-nnf-exlim 37118

Description: Proof of the closed form of exlimi 2231 from modalK (compare exlimiv 1938). See also bj-sylget2 36960. (Contributed by BJ, 2-Dec-2023.)

Assertion

Ref	Expression
bj-nnf-exlim	⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜓)))

Proof of Theorem bj-nnf-exlim

Step	Hyp	Ref	Expression
1		exim 1842	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
2		bj-nnfe 37089	. 2 ⊢ (Ⅎ'𝑥𝜓 → (∃𝑥𝜓 → 𝜓))
3	1, 2	syl9r 78	1 ⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜓)))

Syntax hints:   → wi 4  ∀wal 1546  ∃wex 1787  Ⅎ'wnnf 37084

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 209  df-an 398  df-ex 1788  df-bj-nnf 37085

This theorem is referenced by:  bj-19.23t  37120