![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > 3exdistr | GIF version |
Description: Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
3exdistr | ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anass 928 | . . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒))) | |
2 | 1 | 2exbii 1542 | . . 3 ⊢ (∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑦∃𝑧(𝜑 ∧ (𝜓 ∧ 𝜒))) |
3 | 19.42vv 1836 | . . 3 ⊢ (∃𝑦∃𝑧(𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ (𝜑 ∧ ∃𝑦∃𝑧(𝜓 ∧ 𝜒))) | |
4 | exdistr 1835 | . . . 4 ⊢ (∃𝑦∃𝑧(𝜓 ∧ 𝜒) ↔ ∃𝑦(𝜓 ∧ ∃𝑧𝜒)) | |
5 | 4 | anbi2i 445 | . . 3 ⊢ ((𝜑 ∧ ∃𝑦∃𝑧(𝜓 ∧ 𝜒)) ↔ (𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒))) |
6 | 2, 3, 5 | 3bitri 204 | . 2 ⊢ (∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒))) |
7 | 6 | exbii 1541 | 1 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 ∧ w3a 924 ∃wex 1426 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1381 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-4 1445 ax-17 1464 ax-ial 1472 |
This theorem depends on definitions: df-bi 115 df-3an 926 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |