Theorem rmo2ilem 2993

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 1446	. . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦)))
3		df-ral 2419	. . . 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 1584	. 2 ⊢ (∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))
6		nfv 1508	. . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
7		rmo2.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
8	6, 7	nfan 1544	. . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)
9	8	mo2r 2049	. . 3 ⊢ (∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
10		df-rmo 2422	. . 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 1329   = wceq 1331  Ⅎwnf 1436  ∃wex 1468   ∈ wcel 1480  ∃*wmo 1998  ∀wral 2414  ∃*wrmo 2417

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-ral 2419  df-rmo 2422

This theorem is referenced by:  rmo2i  2994