Theorem bj-19.3t 32689

Description: Closed form of 19.3 2069. (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 2053	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		id 22	. 2 ⊢ ((𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜑))
3	1, 2	impbid2 216	1 ⊢ ((𝜑 → ∀𝑥𝜑) → (∀𝑥𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by: (None)