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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 888 | . . . 4 ⊢ ((φ ∧ ψ ∧ χ) ↔ (φ ∧ (ψ ∧ χ))) | |
2 | 1 | 2exbii 1494 | . . 3 ⊢ (∃y∃z(φ ∧ ψ ∧ χ) ↔ ∃y∃z(φ ∧ (ψ ∧ χ))) |
3 | 19.42vv 1785 | . . 3 ⊢ (∃y∃z(φ ∧ (ψ ∧ χ)) ↔ (φ ∧ ∃y∃z(ψ ∧ χ))) | |
4 | exdistr 1784 | . . . 4 ⊢ (∃y∃z(ψ ∧ χ) ↔ ∃y(ψ ∧ ∃zχ)) | |
5 | 4 | anbi2i 430 | . . 3 ⊢ ((φ ∧ ∃y∃z(ψ ∧ χ)) ↔ (φ ∧ ∃y(ψ ∧ ∃zχ))) |
6 | 2, 3, 5 | 3bitri 195 | . 2 ⊢ (∃y∃z(φ ∧ ψ ∧ χ) ↔ (φ ∧ ∃y(ψ ∧ ∃zχ))) |
7 | 6 | exbii 1493 | 1 ⊢ (∃x∃y∃z(φ ∧ ψ ∧ χ) ↔ ∃x(φ ∧ ∃y(ψ ∧ ∃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-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 |
This theorem is referenced by: (None) |
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