Theorem bj-19.36im 36772

Description: One direction of 19.36 2230 from the same axioms as 19.36imv 1945. (Contributed by BJ, 2-Dec-2023.)

Assertion

Ref	Expression
bj-19.36im	⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)))

Proof of Theorem bj-19.36im

Step	Hyp	Ref	Expression
1		19.35 1877	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2		bj-nnfe 36732	. . 3 ⊢ (Ⅎ'𝑥𝜓 → (∃𝑥𝜓 → 𝜓))
3	2	imim2d 57	. 2 ⊢ (Ⅎ'𝑥𝜓 → ((∀𝑥𝜑 → ∃𝑥𝜓) → (∀𝑥𝜑 → 𝜓)))
4	1, 3	biimtrid 242	1 ⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)))

Syntax hints:   → wi 4  ∀wal 1538  ∃wex 1779  Ⅎ'wnnf 36724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-bj-nnf 36725

This theorem is referenced by: (None)