Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 2451 | . . 3 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} | |
2 | 1 | raleqi 2663 | . 2 ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∀𝑥 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}𝜓) |
3 | ralab2.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) | |
4 | 3 | ralab2 2885 | . 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}𝜓 ↔ ∀𝑦((𝑦 ∈ 𝐴 ∧ 𝜑) → 𝜒)) |
5 | impexp 261 | . . . 4 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝜑) → 𝜒) ↔ (𝑦 ∈ 𝐴 → (𝜑 → 𝜒))) | |
6 | 5 | albii 1457 | . . 3 ⊢ (∀𝑦((𝑦 ∈ 𝐴 ∧ 𝜑) → 𝜒) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝜑 → 𝜒))) |
7 | df-ral 2447 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 (𝜑 → 𝜒) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝜑 → 𝜒))) | |
8 | 6, 7 | bitr4i 186 | . 2 ⊢ (∀𝑦((𝑦 ∈ 𝐴 ∧ 𝜑) → 𝜒) ↔ ∀𝑦 ∈ 𝐴 (𝜑 → 𝜒)) |
9 | 2, 4, 8 | 3bitri 205 | 1 ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∀𝑦 ∈ 𝐴 (𝜑 → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1340 ∈ wcel 2135 {cab 2150 ∀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 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |