Theorem rmoeqALT 29327

Description: Equality's restricted existential "at most one" property. (Contributed by Thierry Arnoux, 30-Mar-2018.) Obsolete version of rmoeq 3405 as of 27-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
rmoeqALT	⊢ (𝐴 ∈ 𝑉 → ∃*𝑥 ∈ 𝐵 𝑥 = 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem rmoeqALT

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
2	1	rgenw 2924	. . 3 ⊢ ∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝑥 = 𝐴)
3		eqeq2 2633	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
4	3	imbi2d 330	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑥 = 𝐴 → 𝑥 = 𝑦) ↔ (𝑥 = 𝐴 → 𝑥 = 𝐴)))
5	4	ralbidv 2986	. . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝑥 = 𝐴)))
6	5	spcegv 3294	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝑥 = 𝐴) → ∃𝑦∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝑥 = 𝑦)))
7	2, 6	mpi 20	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝑥 = 𝑦))
8		nfv 1843	. . 3 ⊢ Ⅎ𝑦 𝑥 = 𝐴
9	8	rmo2 3526	. 2 ⊢ (∃*𝑥 ∈ 𝐵 𝑥 = 𝐴 ↔ ∃𝑦∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝑥 = 𝑦))
10	7, 9	sylibr 224	1 ⊢ (𝐴 ∈ 𝑉 → ∃*𝑥 ∈ 𝐵 𝑥 = 𝐴)

Syntax hints:   → wi 4   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  ∃*wrmo 2915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rmo 2920  df-v 3202

This theorem is referenced by: (None)