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Theorem wl-dfralv 35005
 Description: Alternate definition of restricted universal quantification (df-wl-ral ) when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 23-May-2023.)
Assertion
Ref Expression
wl-dfralv (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem wl-dfralv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-wl-ral 35000 . 2 (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑦(𝑦𝐴 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 19.21v 1940 . . 3 (∀𝑥(𝑦𝐴 → (𝑥 = 𝑦𝜑)) ↔ (𝑦𝐴 → ∀𝑥(𝑥 = 𝑦𝜑)))
32albii 1821 . 2 (∀𝑦𝑥(𝑦𝐴 → (𝑥 = 𝑦𝜑)) ↔ ∀𝑦(𝑦𝐴 → ∀𝑥(𝑥 = 𝑦𝜑)))
4 wl-dfralflem 35002 . 2 (∀𝑦𝑥(𝑦𝐴 → (𝑥 = 𝑦𝜑)) ↔ ∀𝑥(𝑥𝐴𝜑))
51, 3, 43bitr2i 302 1 (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   ∈ wcel 2112  ∀wl-ral 34995 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 2114  ax-11 2159 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-clel 2873  df-wl-ral 35000 This theorem is referenced by:  wl-rgen  35006  wl-dfrexv  35013  wl-dfrmov  35018
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