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Mirrors > Home > ILE Home > Th. List > 19.41vvv | GIF version |
Description: Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 30-Apr-1995.) |
Ref | Expression |
---|---|
19.41vvv | ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.41vv 1857 | . . 3 ⊢ (∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (∃𝑦∃𝑧𝜑 ∧ 𝜓)) | |
2 | 1 | exbii 1567 | . 2 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑥(∃𝑦∃𝑧𝜑 ∧ 𝜓)) |
3 | 19.41v 1856 | . 2 ⊢ (∃𝑥(∃𝑦∃𝑧𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓)) | |
4 | 2, 3 | bitri 183 | 1 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∃wex 1451 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1406 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-4 1470 ax-17 1489 ax-ial 1497 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: 19.41vvvv 1859 eloprabga 5812 dftpos3 6113 |
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