Theorem bj-19.36im 37311

Description: One direction of 19.36 2272 from the same axioms as 19.36imv 1972. (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		bj-nnfe 37279	. 2 ⊢ (Ⅎ'𝑥𝜓 → (∃𝑥𝜓 → 𝜓))
2		bj-spimnfe 37169	. 2 ⊢ ((∃𝑥𝜓 → 𝜓) → (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)))
3	1, 2	syl 18	1 ⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)))

Syntax hints:   → wi 4  ∀wal 1565  ∃wex 1806  Ⅎ'wnnf 37274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-bj-nnf 37275

This theorem is referenced by: (None)