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Mirrors > Home > MPE Home > Th. List > eeeanv | Structured version Visualization version GIF version |
Description: Distribute three existential quantifiers over a conjunction. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) Reduce distinct variable restrictions. (Revised by Wolf Lammen, 20-Jan-2018.) |
Ref | Expression |
---|---|
eeeanv | ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eeanv 2346 | . . 3 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) | |
2 | 1 | anbi1i 625 | . 2 ⊢ ((∃𝑥∃𝑦(𝜑 ∧ 𝜓) ∧ ∃𝑧𝜒) ↔ ((∃𝑥𝜑 ∧ ∃𝑦𝜓) ∧ ∃𝑧𝜒)) |
3 | df-3an 1090 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) | |
4 | 3 | exbii 1851 | . . . . 5 ⊢ (∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑧((𝜑 ∧ 𝜓) ∧ 𝜒)) |
5 | 19.42v 1958 | . . . . 5 ⊢ (∃𝑧((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ ∃𝑧𝜒)) | |
6 | 4, 5 | bitri 275 | . . . 4 ⊢ (∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ ∃𝑧𝜒)) |
7 | 6 | 2exbii 1852 | . . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑥∃𝑦((𝜑 ∧ 𝜓) ∧ ∃𝑧𝜒)) |
8 | nfv 1918 | . . . . . 6 ⊢ Ⅎ𝑦𝜒 | |
9 | 8 | nfex 2318 | . . . . 5 ⊢ Ⅎ𝑦∃𝑧𝜒 |
10 | 9 | 19.41 2229 | . . . 4 ⊢ (∃𝑦((𝜑 ∧ 𝜓) ∧ ∃𝑧𝜒) ↔ (∃𝑦(𝜑 ∧ 𝜓) ∧ ∃𝑧𝜒)) |
11 | 10 | exbii 1851 | . . 3 ⊢ (∃𝑥∃𝑦((𝜑 ∧ 𝜓) ∧ ∃𝑧𝜒) ↔ ∃𝑥(∃𝑦(𝜑 ∧ 𝜓) ∧ ∃𝑧𝜒)) |
12 | nfv 1918 | . . . . 5 ⊢ Ⅎ𝑥𝜒 | |
13 | 12 | nfex 2318 | . . . 4 ⊢ Ⅎ𝑥∃𝑧𝜒 |
14 | 13 | 19.41 2229 | . . 3 ⊢ (∃𝑥(∃𝑦(𝜑 ∧ 𝜓) ∧ ∃𝑧𝜒) ↔ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ∧ ∃𝑧𝜒)) |
15 | 7, 11, 14 | 3bitri 297 | . 2 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ∧ ∃𝑧𝜒)) |
16 | df-3an 1090 | . 2 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒) ↔ ((∃𝑥𝜑 ∧ ∃𝑦𝜓) ∧ ∃𝑧𝜒)) | |
17 | 2, 15, 16 | 3bitr4i 303 | 1 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∃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 1914 ax-6 1972 ax-7 2012 ax-10 2138 ax-11 2155 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-ex 1783 df-nf 1787 |
This theorem is referenced by: eloprabgaOLD 7470 |
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