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Theorem wl-dfralfi 35319
 Description: Restricted universal quantification (df-wl-ral 35315) allows allows a simplification, if we can assume all occurrences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 26-May-2023.)
Hypothesis
Ref Expression
wl-dralfi.1 𝑥𝐴
Assertion
Ref Expression
wl-dfralfi (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))

Proof of Theorem wl-dfralfi
StepHypRef Expression
1 wl-dralfi.1 . 2 𝑥𝐴
2 wl-dfralf 35318 . 2 (𝑥𝐴 → (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑥(𝑥𝐴𝜑)))
31, 2ax-mp 5 1 (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   ∈ wcel 2111  Ⅎwnfc 2899  ∀wl-ral 35310 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-11 2158  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-clel 2830  df-nfc 2901  df-wl-ral 35315 This theorem is referenced by: (None)
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