Theorem wl-dfralv 34843

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(𝑥 ∈ 𝐴 → 𝜑))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem wl-dfralv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-wl-ral 34838	. 2 ⊢ (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		19.21v 1940	. . 3 ⊢ (∀𝑥(𝑦 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) ↔ (𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3	2	albii 1820	. 2 ⊢ (∀𝑦∀𝑥(𝑦 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑦(𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
4		wl-dfralflem 34840	. 2 ⊢ (∀𝑦∀𝑥(𝑦 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
5	1, 3, 4	3bitr2i 301	1 ⊢ (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535   ∈ wcel 2114  ∀wl-ral 34833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-11 2161

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-clel 2895  df-wl-ral 34838

This theorem is referenced by:  wl-rgen  34844  wl-dfrexv  34851  wl-dfrmov  34856