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Theorem r19.3rz 4095
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 3964 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 biimt 349 . . 3 (∃𝑥 𝑥𝐴 → (𝜑 ↔ (∃𝑥 𝑥𝐴𝜑)))
31, 2sylbi 207 . 2 (𝐴 ≠ ∅ → (𝜑 ↔ (∃𝑥 𝑥𝐴𝜑)))
4 df-ral 2946 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
5 r19.3rz.1 . . . 4 𝑥𝜑
6519.23 2118 . . 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 1521  wex 1744  wnf 1748  wcel 2030  wne 2823  wral 2941  c0 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-v 3233  df-dif 3610  df-nul 3949
This theorem is referenced by:  r19.28z  4096  r19.3rzv  4097  r19.27z  4103  2reu4a  41510
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