![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > cbvral3v | GIF version |
Description: Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.) |
Ref | Expression |
---|---|
cbvral3v.1 | ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒)) |
cbvral3v.2 | ⊢ (𝑦 = 𝑣 → (𝜒 ↔ 𝜃)) |
cbvral3v.3 | ⊢ (𝑧 = 𝑢 → (𝜃 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvral3v | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvral3v.1 | . . . 4 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒)) | |
2 | 1 | 2ralbidv 2462 | . . 3 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒)) |
3 | 2 | cbvralv 2657 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒) |
4 | cbvral3v.2 | . . . 4 ⊢ (𝑦 = 𝑣 → (𝜒 ↔ 𝜃)) | |
5 | cbvral3v.3 | . . . 4 ⊢ (𝑧 = 𝑢 → (𝜃 ↔ 𝜓)) | |
6 | 4, 5 | cbvral2v 2668 | . . 3 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒 ↔ ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓) |
7 | 6 | ralbii 2444 | . 2 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒 ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓) |
8 | 3, 7 | bitri 183 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wral 2417 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |