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 1960. (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 2162 | . . 3 ⊢ (∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ ∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓)) | |
2 | 1 | exbii 1850 | . 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓)) |
3 | eeanv 2347 | . . 3 ⊢ (∃𝑦∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑦𝜑 ∧ ∃𝑤𝜓)) | |
4 | 3 | 2exbii 1851 | . 2 ⊢ (∃𝑥∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓)) |
5 | eeanv 2347 | . 2 ⊢ (∃𝑥∃𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓)) | |
6 | 2, 4, 5 | 3bitri 297 | 1 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∃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 ax-6 1971 ax-7 2011 ax-10 2137 ax-11 2154 ax-12 2171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ex 1783 df-nf 1787 |
This theorem is referenced by: 5oalem7 30022 elfuns 34217 |
Copyright terms: Public domain | W3C validator |