Theorem bj-19.37im 37080

Description: One direction of 19.37 2240 from the same axioms as 19.37imv 1949. (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 37044	. 2 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
2		bj-spimenfa 36938	. 2 ⊢ ((𝜑 → ∀𝑥𝜑) → (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)))
3	1, 2	syl 17	1 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)))

Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1781  Ⅎ'wnnf 37042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-bj-nnf 37043

This theorem is referenced by:  bj-nnf-spime  37091