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Mirrors > Home > ILE Home > Th. List > exdistrv | GIF version |
Description: Distribute a pair of existential quantifiers (over disjoint variables) over a conjunction. Combination of 19.41v 1875 and 19.42v 1879. For a version with fewer disjoint variable conditions but requiring more axioms, see eeanv 1905. (Contributed by BJ, 30-Sep-2022.) |
Ref | Expression |
---|---|
exdistrv | ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exdistr 1882 | . 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓)) | |
2 | 19.41v 1875 | . 2 ⊢ (∃𝑥(𝜑 ∧ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) | |
3 | 1, 2 | bitri 183 | 1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∃wex 1469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1488 ax-17 1507 ax-ial 1515 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: prodmodc 11379 txbasval 12475 |
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