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Mirrors > Home > ILE Home > Th. List > exdistr2 | GIF version |
Description: Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.) |
Ref | Expression |
---|---|
exdistr2 | ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦∃𝑧𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.42vv 1788 | . 2 ⊢ (∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦∃𝑧𝜓)) | |
2 | 1 | exbii 1496 | 1 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦∃𝑧𝜓)) |
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: (None) |
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