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Mirrors > Home > ILE Home > Th. List > exdistr | GIF version |
Description: Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.) |
Ref | Expression |
---|---|
exdistr | ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.42v 1878 | . 2 ⊢ (∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓)) | |
2 | 1 | exbii 1584 | 1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∃wex 1468 |
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 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-17 1506 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: exdistrv 1882 19.42vv 1883 3exdistr 1887 sbel2x 1971 sbexyz 1976 sbccomlem 2978 uniuni 4367 coass 5052 subhalfnqq 7215 |
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