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Mirrors > Home > ILE Home > Th. List > 19.42vvvv | GIF version |
Description: Theorem 19.42 of [Margaris] p. 90 with 4 quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.) |
Ref | Expression |
---|---|
19.42vvvv | ⊢ (∃𝑤∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑤∃𝑥∃𝑦∃𝑧𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.42vv 1904 | . . 3 ⊢ (∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦∃𝑧𝜓)) | |
2 | 1 | 2exbii 1599 | . 2 ⊢ (∃𝑤∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑤∃𝑥(𝜑 ∧ ∃𝑦∃𝑧𝜓)) |
3 | 19.42vv 1904 | . 2 ⊢ (∃𝑤∃𝑥(𝜑 ∧ ∃𝑦∃𝑧𝜓) ↔ (𝜑 ∧ ∃𝑤∃𝑥∃𝑦∃𝑧𝜓)) | |
4 | 2, 3 | bitri 183 | 1 ⊢ (∃𝑤∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑤∃𝑥∃𝑦∃𝑧𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∃wex 1485 |
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 1440 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-4 1503 ax-17 1519 ax-ial 1527 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: ceqsex8v 2775 enq0tr 7383 |
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