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Mirrors > Home > MPE Home > Th. List > 2eu7 | Structured version Visualization version GIF version |
Description: Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.) |
Ref | Expression |
---|---|
2eu7 | ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfe1 2145 | . . . 4 ⊢ Ⅎ𝑥∃𝑥𝜑 | |
2 | 1 | nfeu 2673 | . . 3 ⊢ Ⅎ𝑥∃!𝑦∃𝑥𝜑 |
3 | 2 | euan 2699 | . 2 ⊢ (∃!𝑥(∃!𝑦∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦∃𝑥𝜑 ∧ ∃!𝑥∃𝑦𝜑)) |
4 | ancom 461 | . . . . 5 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃𝑦𝜑 ∧ ∃𝑥𝜑)) | |
5 | 4 | eubii 2663 | . . . 4 ⊢ (∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑦(∃𝑦𝜑 ∧ ∃𝑥𝜑)) |
6 | nfe1 2145 | . . . . 5 ⊢ Ⅎ𝑦∃𝑦𝜑 | |
7 | 6 | euan 2699 | . . . 4 ⊢ (∃!𝑦(∃𝑦𝜑 ∧ ∃𝑥𝜑) ↔ (∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)) |
8 | ancom 461 | . . . 4 ⊢ ((∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃!𝑦∃𝑥𝜑 ∧ ∃𝑦𝜑)) | |
9 | 5, 7, 8 | 3bitri 298 | . . 3 ⊢ (∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦∃𝑥𝜑 ∧ ∃𝑦𝜑)) |
10 | 9 | eubii 2663 | . 2 ⊢ (∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥(∃!𝑦∃𝑥𝜑 ∧ ∃𝑦𝜑)) |
11 | ancom 461 | . 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃!𝑦∃𝑥𝜑 ∧ ∃!𝑥∃𝑦𝜑)) | |
12 | 3, 10, 11 | 3bitr4ri 305 | 1 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∃wex 1771 ∃!weu 2646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-11 2151 ax-12 2167 ax-13 2381 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-mo 2615 df-eu 2647 |
This theorem is referenced by: 2eu8 2739 |
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