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Mirrors > Home > ILE Home > Th. List > 19.42vv | GIF version |
Description: Theorem 19.42 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 16-Mar-1995.) |
Ref | Expression |
---|---|
19.42vv | ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exdistr 1828 | . 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓)) | |
2 | 19.42v 1827 | . 2 ⊢ (∃𝑥(𝜑 ∧ ∃𝑦𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦𝜓)) | |
3 | 1, 2 | bitri 182 | 1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 ∃wex 1421 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-17 1459 ax-ial 1467 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: 19.42vvv 1830 19.42vvvv 1831 exdistr2 1832 3exdistr 1833 ceqsex3v 2641 ceqsex4v 2642 elvvv 4421 dfoprab2 5572 resoprab 5617 ovi3 5657 ov6g 5658 oprabex3 5776 xpassen 6327 enq0enq 6621 enq0sym 6622 nqnq0pi 6628 |
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