Theorem 19.23t 2208

Description: Closed form of Theorem 19.23 of [Margaris] p. 90. See 19.23 2209. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) df-nf 1781 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 1838	. 2 ⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))
2		19.3t 2199	. . 3 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥𝜓 ↔ 𝜓))
3	2	imbi2d 340	. 2 ⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
4	1, 3	bitr3d 281	1 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535  ∃wex 1776  Ⅎwnf 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-12 2175

This theorem depends on definitions:  df-bi 207  df-ex 1777  df-nf 1781

This theorem is referenced by:  19.23  2209  axie2  2701  r19.23t  3253  ceqsalt  3513  spcimgft  3546  sbciegftOLD  3830  bj-ceqsalt0  36867  bj-ceqsalt1  36868  wl-equsald  37520  wl-equsaldv  37521