Theorem 19.3t 2209

Description: Closed form of 19.3 2210 and version of 19.9t 2212 with a universal quantifier. (Contributed by NM, 9-Nov-2020.) (Proof shortened by BJ, 9-Oct-2022.)

Assertion

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

Proof of Theorem 19.3t

Step	Hyp	Ref	Expression
1		sp 2191	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		nf5r 2202	. 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
3	1, 2	impbid2 226	1 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540  Ⅎwnf 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-ex 1782  df-nf 1786

This theorem is referenced by:  19.23t  2218