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Mirrors > Home > ILE Home > Th. List > ralrab | GIF version |
Description: Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
ralab.1 | ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ralrab | ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜒 ↔ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralab.1 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) | |
2 | 1 | elrab 2877 | . . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜓)) |
3 | 2 | imbi1i 237 | . . 3 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} → 𝜒) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜒)) |
4 | impexp 261 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜒) ↔ (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) | |
5 | 3, 4 | bitri 183 | . 2 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} → 𝜒) ↔ (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) |
6 | 5 | ralbii2 2474 | 1 ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜒 ↔ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 2135 ∀wral 2442 {crab 2446 |
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 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rab 2451 df-v 2723 |
This theorem is referenced by: limcdifap 13178 |
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