Theorem reu6dv 32866

Description: A condition which implies existential uniqueness. (Contributed by Thierry Arnoux, 13-Oct-2025.)

Hypotheses

Ref	Expression
reu6d.1	⊢ (𝜑 → 𝐵 ∈ 𝐴)
reu6d.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝑥 = 𝐵))

Assertion

Ref	Expression
reu6dv	⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓)

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

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem reu6dv

Step	Hyp	Ref	Expression
1		reu6d.1	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
2		reu6d.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝑥 = 𝐵))
3	2	ralrimiva 3159	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝐵))
4		reu6i 3693	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜓)
5	1, 3, 4	syl2anc 596	1 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃!wreu 3369

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-reu 3372

This theorem is used by:  gsummptrev  33416  gsummptp1  33417  gsummulsubdishift1  33428  elrgspnsubrunlem1  33607  selvply1rhmlemb  33949  evlextv  33972  mplvrpmrhm  33977