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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 2234 | . . 3 ⊢ (∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓)) |
3 | 2 | exbii 1855 | . 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓)) |
4 | eean.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
5 | 4 | nfex 2323 | . . 3 ⊢ Ⅎ𝑥∃𝑦𝜓 |
6 | 5 | 19.41 2233 | . 2 ⊢ (∃𝑥(𝜑 ∧ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
7 | 3, 6 | bitri 278 | 1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∃wex 1787 Ⅎwnf 1791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-10 2141 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ex 1788 df-nf 1792 |
This theorem is referenced by: eeanv 2350 reean 3275 |
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