Theorem reu6dv 32473

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 3125	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝐵))
4		reu6i 3683	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜓)
5	1, 3, 4	syl2anc 584	1 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ∃!wreu 3345

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-reu 3348

This theorem is referenced by:  gsummptrev  33067  gsummptp1  33068  gsummulsubdishift1  33079  elrgspnsubrunlem1  33257  evlextv  33635  mplvrpmrhm  33640