Theorem bj-alrim 31870

Description: Closed form of alrimi 2069. (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 2052	. 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
2		sylgt 1739	. 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 1473  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701

This theorem is referenced by:  bj-alrim2  31871