Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  r19.3rzf Structured version   Visualization version   GIF version

Theorem r19.3rzf 45100
Description: Restricted quantification of wff not containing quantified variable. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
Hypotheses
Ref Expression
r19.3rzf.1 𝑥𝜑
r19.3rzf.2 𝑥𝐴
Assertion
Ref Expression
r19.3rzf (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))

Proof of Theorem r19.3rzf
StepHypRef Expression
1 r19.3rzf.2 . . . 4 𝑥𝐴
21n0f 4354 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
3 biimt 360 . . 3 (∃𝑥 𝑥𝐴 → (𝜑 ↔ (∃𝑥 𝑥𝐴𝜑)))
42, 3sylbi 217 . 2 (𝐴 ≠ ∅ → (𝜑 ↔ (∃𝑥 𝑥𝐴𝜑)))
5 df-ral 3059 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
6 r19.3rzf.1 . . . 4 𝑥𝜑
7619.23 2208 . . 3 (∀𝑥(𝑥𝐴𝜑) ↔ (∃𝑥 𝑥𝐴𝜑))
85, 7bitri 275 . 2 (∀𝑥𝐴 𝜑 ↔ (∃𝑥 𝑥𝐴𝜑))
94, 8bitr4di 289 1 (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1534  wex 1775  wnf 1779  wcel 2105  wnfc 2887  wne 2937  wral 3058  c0 4338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-dif 3965  df-nul 4339
This theorem is referenced by:  r19.28zf  45101
  Copyright terms: Public domain W3C validator