Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ee4anv | Structured version Visualization version GIF version |
Description: Distribute two pairs of existential quantifiers over a conjunction. For a version requiring fewer axioms but with additional disjoint variable conditions, see 4exdistrv 1963. (Contributed by NM, 31-Jul-1995.) |
Ref | Expression |
---|---|
ee4anv | ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | excom 2169 | . . 3 ⊢ (∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ ∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓)) | |
2 | 1 | exbii 1854 | . 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓)) |
3 | eeanv 2351 | . . 3 ⊢ (∃𝑦∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑦𝜑 ∧ ∃𝑤𝜓)) | |
4 | 3 | 2exbii 1855 | . 2 ⊢ (∃𝑥∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓)) |
5 | eeanv 2351 | . 2 ⊢ (∃𝑥∃𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓)) | |
6 | 2, 4, 5 | 3bitri 300 | 1 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∃wex 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-10 2144 ax-11 2161 ax-12 2178 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ex 1787 df-nf 1791 |
This theorem is referenced by: 5oalem7 29587 elfuns 33847 |
Copyright terms: Public domain | W3C validator |