Theorem rmo2i 3844

Description: Condition implying restricted "at most one". (Contributed by NM, 17-Jun-2017.)

Hypothesis

Ref	Expression
rmo2.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
rmo2i	⊢ (∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) → ∃*𝑥 ∈ 𝐴 𝜑)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐴(𝑥)

Proof of Theorem rmo2i

Step	Hyp	Ref	Expression
1		rexex 3098	. 2 ⊢ (∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) → ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))
2		rmo2.1	. . 3 ⊢ Ⅎ𝑦𝜑
3	2	rmo2 3843	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))
4	1, 3	sylibr 237	1 ⊢ (∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) → ∃*𝑥 ∈ 𝐴 𝜑)

This proof depends on syntax axioms:   → wi 4  ∃wex 1812  Ⅎwnf 1816  ∀wral 3082  ∃wrex 3092  ∃*wrmo 3371

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-10 2179  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-mo 2570  df-ral 3083  df-rex 3093  df-rmo 3372

This theorem is used by:  prlngmo2  29243  mndmolinv  42903  modelaxreplem2  45729