Theorem bj-19.36im 37449

Description: One direction of 19.36 2269 from the same axioms as 19.36imv 1978. (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 37417	. 2 ⊢ (Ⅎ'𝑥𝜓 → (∃𝑥𝜓 → 𝜓))
2		bj-spimnfe 37307	. 2 ⊢ ((∃𝑥𝜓 → 𝜓) → (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)))
3	1, 2	syl 18	1 ⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)))

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: (None)