Theorem 19.9t 2241

Description: Closed form of 19.9 2242 and version of 19.3t 2238 with an existential quantifier. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 14-Jul-2020.)

Assertion

Ref	Expression
19.9t	⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))

Proof of Theorem 19.9t

Step	Hyp	Ref	Expression
1		id 23	. . 3 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
2	1	19.9d 2240	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → 𝜑))
3		19.8a 2218	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
4	2, 3	impbid1 228	1 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∃wex 1812  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2213

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817

This theorem is used by:  19.9  2242  19.21t  2243  sbft  2304  bj-cbv3tb  37699  bj-spimtv  37706  bj-equsal1t  37734  bj-19.21t0  37742  19.9dev  43269