Theorem 19.23t 2210

Description: Closed form of Theorem 19.23 of [Margaris] p. 90. See 19.23 2211. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) df-nf 1784 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 1841	. 2 ⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))
2		19.3t 2201	. . 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 1538  ∃wex 1779  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-ex 1780  df-nf 1784

This theorem is referenced by:  19.23  2211  axie2  2703  r19.23t  3255  ceqsalt  3515  spcimgft  3546  sbciegftOLD  3826  bj-ceqsalt0  36885  bj-ceqsalt1  36886  wl-equsald  37540  wl-equsaldv  37541