Theorem bj-alrimhi 32604

Description: An inference associated with sylgt 1749 and bj-exlimh 32602. (Contributed by BJ, 12-May-2019.)

Hypothesis

Ref	Expression
bj-alrimhi.1	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
bj-alrimhi	⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜓))

Proof of Theorem bj-alrimhi

Step	Hyp	Ref	Expression
1		df-nf 1710	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2	1	biimpi 206	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
3		bj-alrimhi.1	. . 3 ⊢ (𝜑 → 𝜓)
4	3	alimi 1739	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥𝜓)
5	2, 4	syl6 35	1 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1481  ∃wex 1704  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-nf 1710

This theorem is referenced by: (None)