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Mirrors > Home > MPE Home > Th. List > eeor | Structured version Visualization version GIF version |
Description: Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.) |
Ref | Expression |
---|---|
eeor.1 | ⊢ Ⅎ𝑦𝜑 |
eeor.2 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
eeor | ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eeor.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | 19.45 2207 | . . 3 ⊢ (∃𝑦(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑦𝜓)) |
3 | 2 | exbii 1833 | . 2 ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ ∃𝑥(𝜑 ∨ ∃𝑦𝜓)) |
4 | eeor.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
5 | 4 | nfex 2308 | . . 3 ⊢ Ⅎ𝑥∃𝑦𝜓 |
6 | 5 | 19.44 2206 | . 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 ∨ wo 842 ∃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-or 843 df-ex 1766 df-nf 1770 |
This theorem is referenced by: (None) |
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