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Mirrors > Home > MPE Home > Th. List > eean | Structured version Visualization version GIF version |
Description: Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Ref | Expression |
---|---|
eean.1 | ⊢ Ⅎ𝑦𝜑 |
eean.2 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
eean | ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eean.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | 19.42 2205 | . . 3 ⊢ (∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓)) |
3 | 2 | exbii 1833 | . 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓)) |
4 | eean.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
5 | 4 | nfex 2308 | . . 3 ⊢ Ⅎ𝑥∃𝑦𝜓 |
6 | 5 | 19.41 2204 | . 2 ⊢ (∃𝑥(𝜑 ∧ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
7 | 3, 6 | bitri 276 | 1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∃wex 1765 Ⅎwnf 1769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-10 2114 ax-11 2128 ax-12 2143 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-ex 1766 df-nf 1770 |
This theorem is referenced by: eeanv 2328 reean 3329 |
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