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Mirrors > Home > MPE Home > Th. List > 19.42vvv | Structured version Visualization version GIF version |
Description: Version of 19.42 2105 with three quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 21-Sep-2011.) |
Ref | Expression |
---|---|
19.42vvv | ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦∃𝑧𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.42vv 1920 | . . 3 ⊢ (∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦∃𝑧𝜓)) | |
2 | 1 | exbii 1774 | . 2 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦∃𝑧𝜓)) |
3 | 19.42v 1918 | . 2 ⊢ (∃𝑥(𝜑 ∧ ∃𝑦∃𝑧𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦∃𝑧𝜓)) | |
4 | 2, 3 | bitri 264 | 1 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦∃𝑧𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 ∃wex 1704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 |
This theorem is referenced by: ceqsex6v 3248 |
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