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Mirrors > Home > ILE Home > Th. List > rmo2ilem | GIF version |
Description: Condition implying restricted at-most-one quantifier. (Contributed by Jim Kingdon, 14-Jul-2018.) |
Ref | Expression |
---|---|
rmo2.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
rmo2ilem | ⊢ (∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) → ∃*𝑥 ∈ 𝐴 𝜑) |
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) |
Colors of variables: wff set class |
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 |
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