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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 1957 and 19.42v 1961. For a version requiring fewer axioms but with additional disjoint variable conditions, see exdistrv 1963. (Contributed by NM, 26-Jul-1995.) |
Ref | Expression |
---|---|
eeanv | ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1921 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1921 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | 1, 2 | eean 2351 | 1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
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 1917 ax-6 1975 ax-7 2020 ax-10 2145 ax-11 2162 ax-12 2179 |
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: eeeanv 2353 ee4anv 2354 |
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