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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 1945 and 19.42v 1949. For a version requiring fewer axioms but with additional disjoint variable conditions, see exdistrv 1951. (Contributed by NM, 26-Jul-1995.) |
Ref | Expression |
---|---|
eeanv | ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1909 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1909 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | 1, 2 | eean 2336 | 1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∃wex 1773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-10 2129 ax-11 2146 ax-12 2163 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ex 1774 df-nf 1778 |
This theorem is referenced by: eeeanv 2338 ee4anv 2339 ttrcltr 9708 |
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