Theorem bj-19.37im 34954

Description: One direction of 19.37 2225 from the same axioms as 19.37imv 1951. (Contributed by BJ, 2-Dec-2023.)

Assertion

Ref	Expression
bj-19.37im	⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)))

Proof of Theorem bj-19.37im

Step	Hyp	Ref	Expression
1		19.35 1880	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2		bj-nnfa 34910	. . 3 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
3	2	imim1d 82	. 2 ⊢ (Ⅎ'𝑥𝜑 → ((∀𝑥𝜑 → ∃𝑥𝜓) → (𝜑 → ∃𝑥𝜓)))
4	1, 3	syl5bi 241	1 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)))

Syntax hints:   → wi 4  ∀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: (None)