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Theorem r19.3rz 4285
 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 4159 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 biimt 352 . . 3 (∃𝑥 𝑥𝐴 → (𝜑 ↔ (∃𝑥 𝑥𝐴𝜑)))
31, 2sylbi 209 . 2 (𝐴 ≠ ∅ → (𝜑 ↔ (∃𝑥 𝑥𝐴𝜑)))
4 df-ral 3095 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
5 r19.3rz.1 . . . 4 𝑥𝜑
6519.23 2197 . . 3 (∀𝑥(𝑥𝐴𝜑) ↔ (∃𝑥 𝑥𝐴𝜑))
74, 6bitri 267 . 2 (∀𝑥𝐴 𝜑 ↔ (∃𝑥 𝑥𝐴𝜑))
83, 7syl6bbr 281 1 (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1599  ∃wex 1823  Ⅎwnf 1827   ∈ wcel 2107   ≠ wne 2969  ∀wral 3090  ∅c0 4141 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-dif 3795  df-nul 4142 This theorem is referenced by:  r19.28z  4286  r19.3rzv  4287  r19.27z  4293  2reu4a  42160
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