Theorem reu6dv 32559

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 3130	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝐵))
4		reu6i 3688	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜓)
5	1, 3, 4	syl2anc 585	1 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃!wreu 3350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-reu 3353

This theorem is referenced by:  gsummptrev  33150  gsummptp1  33151  gsummulsubdishift1  33162  elrgspnsubrunlem1  33341  evlextv  33719  mplvrpmrhm  33724