Theorem 19.23t 2079

Description: Closed form of Theorem 1977.23 of [Margaris] p. 90. See 19.23 2080. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) df-nf 1710 changed. (Revised by Wolf Lammen, 11-Sep-2021.)

Assertion

Ref	Expression
19.23t	⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))

Proof of Theorem 19.23t

Step	Hyp	Ref	Expression
1		nfnt 1782	. . 3 ⊢ (Ⅎ𝑥𝜓 → Ⅎ𝑥 ¬ 𝜓)
2		19.21t 2073	. . 3 ⊢ (Ⅎ𝑥 ¬ 𝜓 → (∀𝑥(¬ 𝜓 → ¬ 𝜑) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑)))
3	1, 2	syl 17	. 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(¬ 𝜓 → ¬ 𝜑) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑)))
4		con34b 306	. . 3 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜓 → ¬ 𝜑))
5	4	albii 1747	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ ∀𝑥(¬ 𝜓 → ¬ 𝜑))
6		eximal 1707	. 2 ⊢ ((∃𝑥𝜑 → 𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))
7	3, 5, 6	3bitr4g 303	1 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))

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:  19.23  2080  axie2  2597  r19.23t  3021  ceqsalt  3228  vtoclgft  3254  vtoclgftOLD  3255  sbciegft  3466  bj-ceqsalt0  32873  bj-ceqsalt1  32874  wl-equsald  33325