Theorem wl-dfrexv 34864

Description: Alternate definition of restricted universal quantification (df-wl-rex 34861) when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 25-May-2023.)

Assertion

Ref	Expression
wl-dfrexv	⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem wl-dfrexv

Step	Hyp	Ref	Expression
1		wl-dfralv 34856	. . 3 ⊢ (∀(𝑥 : 𝐴) ¬ 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑))
2	1	notbii 322	. 2 ⊢ (¬ ∀(𝑥 : 𝐴) ¬ 𝜑 ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑))
3		df-wl-rex 34861	. 2 ⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ¬ ∀(𝑥 : 𝐴) ¬ 𝜑)
4		exnalimn 1844	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑))
5	2, 3, 4	3bitr4i 305	1 ⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535  ∃wex 1780   ∈ wcel 2114  ∀wl-ral 34846  ∃wl-rex 34847

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 2893  df-wl-ral 34851  df-wl-rex 34861

This theorem is referenced by:  wl-dfreuv  34873