Theorem rmo2ilem 3035

Description: Condition implying restricted at-most-one quantifier. (Contributed by Jim Kingdon, 14-Jul-2018.)

Hypothesis

Ref	Expression
rmo2.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
rmo2ilem	⊢ (∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) → ∃*𝑥 ∈ 𝐴 𝜑)

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem rmo2ilem

Step	Hyp	Ref	Expression
1		impexp 261	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦)))
2	1	albii 1457	. . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦)))
3		df-ral 2447	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦)))
4	2, 3	bitr4i 186	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))
5	4	exbii 1592	. 2 ⊢ (∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))
6		nfv 1515	. . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
7		rmo2.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
8	6, 7	nfan 1552	. . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)
9	8	mo2r 2065	. . 3 ⊢ (∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
10		df-rmo 2450	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
11	9, 10	sylibr 133	. 2 ⊢ (∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) → ∃*𝑥 ∈ 𝐴 𝜑)
12	5, 11	sylbir 134	1 ⊢ (∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) → ∃*𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1340   = wceq 1342  Ⅎwnf 1447  ∃wex 1479  ∃*wmo 2014   ∈ wcel 2135  ∀wral 2442  ∃*wrmo 2445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522

This theorem depends on definitions:  df-bi 116  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-ral 2447  df-rmo 2450

This theorem is referenced by:  rmo2i  3036