Theorem wl-dfrmov 34398

Description: Alternate definition of restricted "at most one" (df-wl-rmo 34396) when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 28-May-2023.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem wl-dfrmov

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wl-dfralv 34385	. . . 4 ⊢ (∀(𝑥 : 𝐴)(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦)))
2		impexp 451	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦)))
3	2	albii 1802	. . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦)))
4	1, 3	bitr4i 279	. . 3 ⊢ (∀(𝑥 : 𝐴)(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦))
5	4	exbii 1830	. 2 ⊢ (∃𝑦∀(𝑥 : 𝐴)(𝜑 → 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦))
6		df-wl-rmo 34396	. 2 ⊢ (∃*(𝑥 : 𝐴)𝜑 ↔ ∃𝑦∀(𝑥 : 𝐴)(𝜑 → 𝑥 = 𝑦))
7		df-mo 2575	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦))
8	5, 6, 7	3bitr4i 304	1 ⊢ (∃*(𝑥 : 𝐴)𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1520  ∃wex 1762   ∈ wcel 2080  ∃*wmo 2573  ∀wl-ral 34375  ∃*wl-rmo 34377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-11 2125

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1763  df-mo 2575  df-clel 2862  df-wl-ral 34380  df-wl-rmo 34396

This theorem is referenced by:  wl-dfreuv  34402