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Mirrors > Home > MPE Home > Th. List > 3exdistr | Structured version Visualization version GIF version |
Description: Distribution of existential quantifiers in a triple conjunction. (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 1094 | . . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒))) | |
2 | 1 | 2exbii 1851 | . . 3 ⊢ (∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑦∃𝑧(𝜑 ∧ (𝜓 ∧ 𝜒))) |
3 | 19.42vv 1961 | . . 3 ⊢ (∃𝑦∃𝑧(𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ (𝜑 ∧ ∃𝑦∃𝑧(𝜓 ∧ 𝜒))) | |
4 | exdistr 1958 | . . . 4 ⊢ (∃𝑦∃𝑧(𝜓 ∧ 𝜒) ↔ ∃𝑦(𝜓 ∧ ∃𝑧𝜒)) | |
5 | 4 | anbi2i 623 | . . 3 ⊢ ((𝜑 ∧ ∃𝑦∃𝑧(𝜓 ∧ 𝜒)) ↔ (𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒))) |
6 | 2, 3, 5 | 3bitri 297 | . 2 ⊢ (∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒))) |
7 | 6 | exbii 1850 | 1 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∧ w3a 1086 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 |
This theorem depends on definitions: df-bi 206 df-an 397 df-3an 1088 df-ex 1783 |
This theorem is referenced by: 4exdistr 1965 eloprabga 7372 |
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