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Mirrors > Home > MPE Home > Th. List > 2euexv | Structured version Visualization version GIF version |
Description: Double quantification with existential uniqueness. Version of 2euex 2725 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2389. (Contributed by NM, 3-Dec-2001.) (Revised by Wolf Lammen, 2-Oct-2023.) |
Ref | Expression |
---|---|
2euexv | ⊢ (∃!𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eu 2653 | . 2 ⊢ (∃!𝑥∃𝑦𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑)) | |
2 | excom 2168 | . . . 4 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑) | |
3 | nfe1 2153 | . . . . . 6 ⊢ Ⅎ𝑦∃𝑦𝜑 | |
4 | 3 | nfmov 2643 | . . . . 5 ⊢ Ⅎ𝑦∃*𝑥∃𝑦𝜑 |
5 | 19.8a 2179 | . . . . . . 7 ⊢ (𝜑 → ∃𝑦𝜑) | |
6 | 5 | moimi 2626 | . . . . . 6 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥𝜑) |
7 | moeu 2667 | . . . . . 6 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
8 | 6, 7 | sylib 220 | . . . . 5 ⊢ (∃*𝑥∃𝑦𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑)) |
9 | 4, 8 | eximd 2215 | . . . 4 ⊢ (∃*𝑥∃𝑦𝜑 → (∃𝑦∃𝑥𝜑 → ∃𝑦∃!𝑥𝜑)) |
10 | 2, 9 | syl5bi 244 | . . 3 ⊢ (∃*𝑥∃𝑦𝜑 → (∃𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑)) |
11 | 10 | impcom 410 | . 2 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) → ∃𝑦∃!𝑥𝜑) |
12 | 1, 11 | sylbi 219 | 1 ⊢ (∃!𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∃wex 1779 ∃*wmo 2619 ∃!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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-mo 2621 df-eu 2653 |
This theorem is referenced by: 2exeuv 2716 |
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