Theorem bj-19.3t 31876

Description: Closed form of 19.3 2057. (Contributed by BJ, 20-Oct-2019.)

Assertion

Ref	Expression
bj-19.3t	⊢ ((𝜑 → ∀𝑥𝜑) → (∀𝑥𝜑 ↔ 𝜑))

Proof of Theorem bj-19.3t

Step	Hyp	Ref	Expression
1		sp 2041	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		id 22	. 2 ⊢ ((𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜑))
3	1, 2	impbid2 215	1 ⊢ ((𝜑 → ∀𝑥𝜑) → (∀𝑥𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473

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

This theorem is referenced by: (None)