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Mirrors > Home > ILE Home > Th. List > eeeanv | GIF version |
Description: Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
eeeanv | ⊢ (∃x∃y∃z(φ ∧ ψ ∧ χ) ↔ (∃xφ ∧ ∃yψ ∧ ∃zχ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3an 886 | . . 3 ⊢ ((φ ∧ ψ ∧ χ) ↔ ((φ ∧ ψ) ∧ χ)) | |
2 | 1 | 3exbii 1495 | . 2 ⊢ (∃x∃y∃z(φ ∧ ψ ∧ χ) ↔ ∃x∃y∃z((φ ∧ ψ) ∧ χ)) |
3 | eeanv 1804 | . . 3 ⊢ (∃y∃z((φ ∧ ψ) ∧ χ) ↔ (∃y(φ ∧ ψ) ∧ ∃zχ)) | |
4 | 3 | exbii 1493 | . 2 ⊢ (∃x∃y∃z((φ ∧ ψ) ∧ χ) ↔ ∃x(∃y(φ ∧ ψ) ∧ ∃zχ)) |
5 | eeanv 1804 | . . . 4 ⊢ (∃x∃y(φ ∧ ψ) ↔ (∃xφ ∧ ∃yψ)) | |
6 | 5 | anbi1i 431 | . . 3 ⊢ ((∃x∃y(φ ∧ ψ) ∧ ∃zχ) ↔ ((∃xφ ∧ ∃yψ) ∧ ∃zχ)) |
7 | 19.41v 1779 | . . 3 ⊢ (∃x(∃y(φ ∧ ψ) ∧ ∃zχ) ↔ (∃x∃y(φ ∧ ψ) ∧ ∃zχ)) | |
8 | df-3an 886 | . . 3 ⊢ ((∃xφ ∧ ∃yψ ∧ ∃zχ) ↔ ((∃xφ ∧ ∃yψ) ∧ ∃zχ)) | |
9 | 6, 7, 8 | 3bitr4i 201 | . 2 ⊢ (∃x(∃y(φ ∧ ψ) ∧ ∃zχ) ↔ (∃xφ ∧ ∃yψ ∧ ∃zχ)) |
10 | 2, 4, 9 | 3bitri 195 | 1 ⊢ (∃x∃y∃z(φ ∧ ψ ∧ χ) ↔ (∃xφ ∧ ∃yψ ∧ ∃zχ)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∧ w3a 884 ∃wex 1378 |
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-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-17 1416 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-nf 1347 |
This theorem is referenced by: vtocl3 2604 spc3egv 2638 spc3gv 2639 eloprabga 5533 prarloc 6486 |
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