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Mirrors > Home > MPE Home > Th. List > 2euex | Structured version Visualization version GIF version |
Description: Double quantification with existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2367. Use the weaker 2euexv 2623 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2euex | ⊢ (∃!𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eu 2559 | . 2 ⊢ (∃!𝑥∃𝑦𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑)) | |
2 | excom 2152 | . . . 4 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑) | |
3 | nfe1 2140 | . . . . . 6 ⊢ Ⅎ𝑦∃𝑦𝜑 | |
4 | 3 | nfmo 2552 | . . . . 5 ⊢ Ⅎ𝑦∃*𝑥∃𝑦𝜑 |
5 | 19.8a 2170 | . . . . . . 7 ⊢ (𝜑 → ∃𝑦𝜑) | |
6 | 5 | moimi 2535 | . . . . . 6 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥𝜑) |
7 | moeu 2573 | . . . . . 6 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
8 | 6, 7 | sylib 217 | . . . . 5 ⊢ (∃*𝑥∃𝑦𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑)) |
9 | 4, 8 | eximd 2205 | . . . 4 ⊢ (∃*𝑥∃𝑦𝜑 → (∃𝑦∃𝑥𝜑 → ∃𝑦∃!𝑥𝜑)) |
10 | 2, 9 | biimtrid 241 | . . 3 ⊢ (∃*𝑥∃𝑦𝜑 → (∃𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑)) |
11 | 10 | impcom 407 | . 2 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) → ∃𝑦∃!𝑥𝜑) |
12 | 1, 11 | sylbi 216 | 1 ⊢ (∃!𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∃wex 1774 ∃*wmo 2528 ∃!weu 2558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-10 2130 ax-11 2147 ax-12 2167 ax-13 2367 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-nf 1779 df-mo 2530 df-eu 2559 |
This theorem is referenced by: 2exeu 2638 |
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