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Theorem r19.3rzf 44409
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 4337 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
3 biimt 360 . . 3 (∃𝑥 𝑥𝐴 → (𝜑 ↔ (∃𝑥 𝑥𝐴𝜑)))
42, 3sylbi 216 . 2 (𝐴 ≠ ∅ → (𝜑 ↔ (∃𝑥 𝑥𝐴𝜑)))
5 df-ral 3056 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
6 r19.3rzf.1 . . . 4 𝑥𝜑
7619.23 2196 . . 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 205  wal 1531  wex 1773  wnf 1777  wcel 2098  wnfc 2877  wne 2934  wral 3055  c0 4317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-dif 3946  df-nul 4318
This theorem is referenced by:  r19.28zf  44410
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