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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 1949 and 19.42v 1953. For a version requiring fewer axioms but with additional disjoint variable conditions, see exdistrv 1955. (Contributed by NM, 26-Jul-1995.) |
| Ref | Expression |
|---|---|
| eeanv | ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfv 1914 | . 2 ⊢ Ⅎ𝑦𝜑 | |
| 2 | nfv 1914 | . 2 ⊢ Ⅎ𝑥𝜓 | |
| 3 | 1, 2 | eean 2350 | 1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∃wex 1779 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-11 2157 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: eeeanv 2352 ee4anv 2353 ee4anvOLD 2354 ttrcltr 9756 |
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