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Mirrors > Home > ILE Home > Th. List > ralab | GIF version |
Description: Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
ralab.1 | ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ralab | ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 2311 | . 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜒)) | |
2 | vex 2560 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | ralab.1 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | elab 2687 | . . . 4 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝜓) |
5 | 4 | imbi1i 227 | . . 3 ⊢ ((𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜒) ↔ (𝜓 → 𝜒)) |
6 | 5 | albii 1359 | . 2 ⊢ (∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜒) ↔ ∀𝑥(𝜓 → 𝜒)) |
7 | 1, 6 | bitri 173 | 1 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝜓 → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1241 ∈ wcel 1393 {cab 2026 ∀wral 2306 |
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-v 2559 |
This theorem is referenced by: funcnvuni 4968 ralrnmpt2 5615 pitonn 6924 |
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