Theorem bj-alrim 36659

Description: Closed form of alrimi 2214. (Contributed by BJ, 2-May-2019.)

Assertion

Ref	Expression
bj-alrim	⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))

Proof of Theorem bj-alrim

Step	Hyp	Ref	Expression
1		nf5r 2195	. 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
2		sylgt 1820	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜓)))
3	1, 2	syl5com 31	1 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))

Syntax hints:   → wi 4  ∀wal 1535  Ⅎwnf 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-ex 1778  df-nf 1782

This theorem is referenced by:  bj-alrim2  36660