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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 1953 and 19.42v 1957. For a version requiring fewer axioms but with additional disjoint variable conditions, see exdistrv 1959. (Contributed by NM, 26-Jul-1995.) |
Ref | Expression |
---|---|
eeanv | ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1917 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1917 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | 1, 2 | eean 2346 | 1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-11 2154 ax-12 2171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ex 1783 df-nf 1787 |
This theorem is referenced by: eeeanv 2348 ee4anv 2349 ttrcltr 9474 |
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