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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 1969 and 19.42v 1973. For a version requiring fewer axioms but with additional disjoint variable conditions, see exdistrv 1975. (Contributed by NM, 26-Jul-1995.) |
| Ref | Expression |
|---|---|
| eeanv | ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfv 1934 | . 2 ⊢ Ⅎ𝑦𝜑 | |
| 2 | nfv 1934 | . 2 ⊢ Ⅎ𝑥𝜓 | |
| 3 | 1, 2 | eean 2379 | 1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 208 ∧ wa 399 ∃wex 1799 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-10 2175 ax-11 2191 ax-12 2212 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-ex 1800 df-nf 1804 |
| This theorem is referenced by: eeeanv 2381 ee4anv 2382 ee4anvOLD 2383 ttrcltr 9671 |
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