Theorem 19.9alt 33270

Description: Version of 19.9t 2059 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 1767	. . . 4 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)
2		19.9t 2059	. . . 4 ⊢ (Ⅎ𝑥 ¬ 𝜑 → (∃𝑥 ¬ 𝜑 ↔ ¬ 𝜑))
3	1, 2	syl 17	. . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥 ¬ 𝜑 ↔ ¬ 𝜑))
4	3	con2bid 343	. 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑))
5		alex 1743	. 2 ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
6	4, 5	syl6rbbr 278	1 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥𝜑 ↔ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699

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-or 384  df-ex 1696  df-nf 1701

This theorem is referenced by: (None)