Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > eeanv | Structured version Visualization version GIF version |
Description: Distribute a pair of existential quantifiers over a conjunction. Combination of 19.41v 1941 and 19.42v 1945. For a version requiring fewer axioms but with additional disjoint variable conditions, see exdistrv 1947. (Contributed by NM, 26-Jul-1995.) |
Ref | Expression |
---|---|
eeanv | ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1906 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1906 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | 1, 2 | eean 2360 | 1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∃wex 1771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-11 2151 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-nf 1776 |
This theorem is referenced by: eeeanv 2362 ee4anv 2363 |
Copyright terms: Public domain | W3C validator |