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Theorem r19.3rz 4040
Description: Restricted quantification of wff not containing quantified variable. (Contributed by FL, 3-Jan-2008.)
Hypothesis
Ref Expression
r19.3rz.1 𝑥𝜑
Assertion
Ref Expression
r19.3rz (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem r19.3rz
StepHypRef Expression
1 n0 3913 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 biimt 350 . . 3 (∃𝑥 𝑥𝐴 → (𝜑 ↔ (∃𝑥 𝑥𝐴𝜑)))
31, 2sylbi 207 . 2 (𝐴 ≠ ∅ → (𝜑 ↔ (∃𝑥 𝑥𝐴𝜑)))
4 df-ral 2913 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
5 r19.3rz.1 . . . 4 𝑥𝜑
6519.23 2078 . . 3 (∀𝑥(𝑥𝐴𝜑) ↔ (∃𝑥 𝑥𝐴𝜑))
74, 6bitri 264 . 2 (∀𝑥𝐴 𝜑 ↔ (∃𝑥 𝑥𝐴𝜑))
83, 7syl6bbr 278 1 (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478  wex 1701  wnf 1705  wcel 1987  wne 2790  wral 2908  c0 3897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-v 3192  df-dif 3563  df-nul 3898
This theorem is referenced by:  r19.28z  4041  r19.3rzv  4042  r19.27z  4048  2reu4a  40523
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