Theorem 19.23t 2222

Description: Closed form of Theorem 19.23 of [Margaris] p. 90. See 19.23 2223. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) df-nf 1791 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by BJ, 8-Oct-2022.)

Assertion

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

Proof of Theorem 19.23t

Step	Hyp	Ref	Expression
1		19.38b 1848	. 2 ⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))
2		19.3t 2213	. . 3 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥𝜓 ↔ 𝜓))
3	2	imbi2d 341	. 2 ⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
4	1, 3	bitr3d 282	1 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1545  ∃wex 1786  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-ex 1787  df-nf 1791

This theorem is referenced by:  19.23  2223  axie2  2706  r19.23t  3235  ceqsalt  3464  spcimgft  3492  bj-ceqsalt0  37237  bj-ceqsalt1  37238  wl-equsald  37910  wl-equsaldv  37911