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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 1786 | . 2 ⊢ (∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓)) | |
2 | 1 | exbii 1496 | 1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∃wex 1381 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-17 1419 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: 19.42vv 1788 3exdistr 1792 sbel2x 1874 sbexyz 1879 sbccomlem 2832 uniuni 4183 coass 4839 subhalfnqq 6512 |
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