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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 1922 | . 2 ⊢ Ⅎ𝑦𝜑 | |
| 2 | nfv 1922 | . 2 ⊢ Ⅎ𝑥𝜓 | |
| 3 | 1, 2 | eean 2358 | 1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 208 ∧ wa 397 ∃wex 1787 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-10 2154 ax-11 2170 ax-12 2191 |
| This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-ex 1788 df-nf 1792 |
| This theorem is referenced by: eeeanv 2360 ee4anv 2361 ee4anvOLD 2362 ttrcltr 9632 |
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