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Mirrors > Home > MPE Home > Th. List > 2eu8 | Structured version Visualization version GIF version |
Description: Two equivalent expressions for double existential uniqueness. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!𝑥∃!𝑦 using 2eu7 2744. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by NM, 20-Feb-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2eu8 | ⊢ (∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2eu2 2737 | . . 3 ⊢ (∃!𝑥∃𝑦𝜑 → (∃!𝑦∃!𝑥𝜑 ↔ ∃!𝑦∃𝑥𝜑)) | |
2 | 1 | pm5.32i 577 | . 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)) |
3 | nfeu1 2673 | . . . . 5 ⊢ Ⅎ𝑥∃!𝑥𝜑 | |
4 | 3 | nfeu 2679 | . . . 4 ⊢ Ⅎ𝑥∃!𝑦∃!𝑥𝜑 |
5 | 4 | euan 2705 | . . 3 ⊢ (∃!𝑥(∃!𝑦∃!𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦∃!𝑥𝜑 ∧ ∃!𝑥∃𝑦𝜑)) |
6 | ancom 463 | . . . . . 6 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃𝑦𝜑 ∧ ∃!𝑥𝜑)) | |
7 | 6 | eubii 2669 | . . . . 5 ⊢ (∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑦(∃𝑦𝜑 ∧ ∃!𝑥𝜑)) |
8 | nfe1 2153 | . . . . . 6 ⊢ Ⅎ𝑦∃𝑦𝜑 | |
9 | 8 | euan 2705 | . . . . 5 ⊢ (∃!𝑦(∃𝑦𝜑 ∧ ∃!𝑥𝜑) ↔ (∃𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑)) |
10 | ancom 463 | . . . . 5 ⊢ ((∃𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑦∃!𝑥𝜑 ∧ ∃𝑦𝜑)) | |
11 | 7, 9, 10 | 3bitri 299 | . . . 4 ⊢ (∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦∃!𝑥𝜑 ∧ ∃𝑦𝜑)) |
12 | 11 | eubii 2669 | . . 3 ⊢ (∃!𝑥∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥(∃!𝑦∃!𝑥𝜑 ∧ ∃𝑦𝜑)) |
13 | ancom 463 | . . 3 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑦∃!𝑥𝜑 ∧ ∃!𝑥∃𝑦𝜑)) | |
14 | 5, 12, 13 | 3bitr4ri 306 | . 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑)) |
15 | 2eu7 2744 | . 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑)) | |
16 | 2, 14, 15 | 3bitr3ri 304 | 1 ⊢ (∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∃wex 1779 ∃!weu 2652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-10 2144 ax-11 2160 ax-12 2176 ax-13 2389 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 |
This theorem is referenced by: (None) |
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