Theorem rmo2ilem 3092

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 263	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦)))
2	1	albii 1494	. . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦)))
3		df-ral 2490	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦)))
4	2, 3	bitr4i 187	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))
5	4	exbii 1629	. 2 ⊢ (∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))
6		nfv 1552	. . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
7		rmo2.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
8	6, 7	nfan 1589	. . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)
9	8	mo2r 2107	. . 3 ⊢ (∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
10		df-rmo 2493	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
11	9, 10	sylibr 134	. 2 ⊢ (∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) → ∃*𝑥 ∈ 𝐴 𝜑)
12	5, 11	sylbir 135	1 ⊢ (∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) → ∃*𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1371   = wceq 1373  Ⅎwnf 1484  ∃wex 1516  ∃*wmo 2056   ∈ wcel 2177  ∀wral 2485  ∃*wrmo 2488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-ral 2490  df-rmo 2493

This theorem is referenced by:  rmo2i  3093