Theorem bj-19.37im 37450

Description: One direction of 19.37 2271 from the same axioms as 19.37imv 1980. (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		bj-nnfa 37414	. 2 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
2		bj-spimenfa 37308	. 2 ⊢ ((𝜑 → ∀𝑥𝜑) → (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)))
3	1, 2	syl 18	1 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812  Ⅎ'wnnf 37412

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-bj-nnf 37413

This theorem is used by:  bj-nnf-spime  37461