Theorem wl-dfralv 34319

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 34314	. 2 ⊢ (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		19.21v 1899	. . 3 ⊢ (∀𝑥(𝑦 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) ↔ (𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3	2	albii 1783	. 2 ⊢ (∀𝑦∀𝑥(𝑦 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑦(𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
4		wl-dfralflem 34316	. 2 ⊢ (∀𝑦∀𝑥(𝑦 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
5	1, 3, 4	3bitr2i 291	1 ⊢ (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1506   ∈ wcel 2051  ∀wl-ral 34309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-11 2094

This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1744  df-clel 2839  df-wl-ral 34314

This theorem is referenced by:  wl-rgen  34320  wl-dfrexv  34327  wl-dfrmov  34332