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Mirrors > Home > ILE Home > Th. List > ralrab2 | GIF version |
Description: Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
ralab2.1 | ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
ralrab2 | ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∀𝑦 ∈ 𝐴 (𝜑 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 2315 | . . 3 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} | |
2 | 1 | raleqi 2509 | . 2 ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∀𝑥 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}𝜓) |
3 | ralab2.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) | |
4 | 3 | ralab2 2705 | . 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}𝜓 ↔ ∀𝑦((𝑦 ∈ 𝐴 ∧ 𝜑) → 𝜒)) |
5 | impexp 250 | . . . 4 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝜑) → 𝜒) ↔ (𝑦 ∈ 𝐴 → (𝜑 → 𝜒))) | |
6 | 5 | albii 1359 | . . 3 ⊢ (∀𝑦((𝑦 ∈ 𝐴 ∧ 𝜑) → 𝜒) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝜑 → 𝜒))) |
7 | df-ral 2311 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 (𝜑 → 𝜒) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝜑 → 𝜒))) | |
8 | 6, 7 | bitr4i 176 | . 2 ⊢ (∀𝑦((𝑦 ∈ 𝐴 ∧ 𝜑) → 𝜒) ↔ ∀𝑦 ∈ 𝐴 (𝜑 → 𝜒)) |
9 | 2, 4, 8 | 3bitri 195 | 1 ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∀𝑦 ∈ 𝐴 (𝜑 → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1241 ∈ wcel 1393 {cab 2026 ∀wral 2306 {crab 2310 |
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-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rab 2315 |
This theorem is referenced by: (None) |
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