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Theorem r19.3rzf 45163
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 4349 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
3 biimt 360 . . 3 (∃𝑥 𝑥𝐴 → (𝜑 ↔ (∃𝑥 𝑥𝐴𝜑)))
42, 3sylbi 217 . 2 (𝐴 ≠ ∅ → (𝜑 ↔ (∃𝑥 𝑥𝐴𝜑)))
5 df-ral 3062 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
6 r19.3rzf.1 . . . 4 𝑥𝜑
7619.23 2211 . . 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 1538  wex 1779  wnf 1783  wcel 2108  wnfc 2890  wne 2940  wral 3061  c0 4333
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-8 2110  ax-9 2118  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-dif 3954  df-nul 4334
This theorem is referenced by:  r19.28zf  45164
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