Theorem wl-dfrexex 34239

Description: Alternate definition of the restricted existential quantification (df-wl-rex 34235), according to the pattern given in df-wl-ral 34225. (Contributed by Wolf Lammen, 25-May-2023.)

Assertion

Ref	Expression
wl-dfrexex	⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem wl-dfrexex

Step	Hyp	Ref	Expression
1		df-wl-ral 34225	. . 3 ⊢ (∀(𝑥 : 𝐴) ¬ 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)))
2	1	notbii 312	. 2 ⊢ (¬ ∀(𝑥 : 𝐴) ¬ 𝜑 ↔ ¬ ∀𝑦(𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)))
3		df-wl-rex 34235	. 2 ⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ¬ ∀(𝑥 : 𝐴) ¬ 𝜑)
4		alinexa 1805	. . . . . 6 ⊢ (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
5	4	imbi2i 328	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) ↔ (𝑦 ∈ 𝐴 → ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
6	5	albii 1782	. . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) ↔ ∀𝑦(𝑦 ∈ 𝐴 → ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
7		alinexa 1805	. . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ↔ ¬ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
8	6, 7	bitri 267	. . 3 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) ↔ ¬ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
9	8	con2bii 350	. 2 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ↔ ¬ ∀𝑦(𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)))
10	2, 3, 9	3bitr4i 295	1 ⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387  ∀wal 1505  ∃wex 1742   ∈ wcel 2048  ∀wl-ral 34220  ∃wl-rex 34221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772

This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-wl-ral 34225  df-wl-rex 34235

This theorem is referenced by: (None)