Theorem 19.9alt 34252

Description: Version of 19.9t 2071 for universal quantifier. (Contributed by NM, 9-Nov-2020.)

Assertion

Ref	Expression
19.9alt	⊢ (Ⅎ𝑥𝜑 → (∀𝑥𝜑 ↔ 𝜑))

Proof of Theorem 19.9alt

Step	Hyp	Ref	Expression
1		nfnt 1782	. . . 4 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)
2		19.9t 2071	. . . 4 ⊢ (Ⅎ𝑥 ¬ 𝜑 → (∃𝑥 ¬ 𝜑 ↔ ¬ 𝜑))
3	1, 2	syl 17	. . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥 ¬ 𝜑 ↔ ¬ 𝜑))
4	3	con2bid 344	. 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑))
5		alex 1753	. 2 ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
6	4, 5	syl6rbbr 279	1 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥𝜑 ↔ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196  ∀wal 1481  ∃wex 1704  Ⅎwnf 1708

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-or 385  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)