Theorem wl-dfrmov 34869

Description: Alternate definition of restricted "at most one" (df-wl-rmo 34867) 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 34856	. . . 4 ⊢ (∀(𝑥 : 𝐴)(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦)))
2		impexp 453	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦)))
3	2	albii 1820	. . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦)))
4	1, 3	bitr4i 280	. . 3 ⊢ (∀(𝑥 : 𝐴)(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦))
5	4	exbii 1848	. 2 ⊢ (∃𝑦∀(𝑥 : 𝐴)(𝜑 → 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦))
6		df-wl-rmo 34867	. 2 ⊢ (∃*(𝑥 : 𝐴)𝜑 ↔ ∃𝑦∀(𝑥 : 𝐴)(𝜑 → 𝑥 = 𝑦))
7		df-mo 2622	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦))
8	5, 6, 7	3bitr4i 305	1 ⊢ (∃*(𝑥 : 𝐴)𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535  ∃wex 1780   ∈ wcel 2114  ∃*wmo 2620  ∀wl-ral 34846  ∃*wl-rmo 34848

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-mo 2622  df-clel 2893  df-wl-ral 34851  df-wl-rmo 34867

This theorem is referenced by:  wl-dfreuv  34873