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Mirrors > Home > ILE Home > Th. List > ralss | GIF version |
Description: Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
Ref | Expression |
---|---|
ralss | ⊢ (A ⊆ B → (∀x ∈ A φ ↔ ∀x ∈ B (x ∈ A → φ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 2933 | . . . . 5 ⊢ (A ⊆ B → (x ∈ A → x ∈ B)) | |
2 | 1 | pm4.71rd 374 | . . . 4 ⊢ (A ⊆ B → (x ∈ A ↔ (x ∈ B ∧ x ∈ A))) |
3 | 2 | imbi1d 220 | . . 3 ⊢ (A ⊆ B → ((x ∈ A → φ) ↔ ((x ∈ B ∧ x ∈ A) → φ))) |
4 | impexp 250 | . . 3 ⊢ (((x ∈ B ∧ x ∈ A) → φ) ↔ (x ∈ B → (x ∈ A → φ))) | |
5 | 3, 4 | syl6bb 185 | . 2 ⊢ (A ⊆ B → ((x ∈ A → φ) ↔ (x ∈ B → (x ∈ A → φ)))) |
6 | 5 | ralbidv2 2322 | 1 ⊢ (A ⊆ B → (∀x ∈ A φ ↔ ∀x ∈ B (x ∈ A → φ))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∈ wcel 1390 ∀wral 2300 ⊆ wss 2911 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-ral 2305 df-in 2918 df-ss 2925 |
This theorem is referenced by: (None) |
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