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Mirrors > Home > MPE Home > Th. List > 2euex | Structured version Visualization version GIF version |
Description: Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
2euex | ⊢ (∃!𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eu5 2484 | . 2 ⊢ (∃!𝑥∃𝑦𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑)) | |
2 | excom 2029 | . . . 4 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑) | |
3 | nfe1 2014 | . . . . . 6 ⊢ Ⅎ𝑦∃𝑦𝜑 | |
4 | 3 | nfmo 2475 | . . . . 5 ⊢ Ⅎ𝑦∃*𝑥∃𝑦𝜑 |
5 | 19.8a 2039 | . . . . . . 7 ⊢ (𝜑 → ∃𝑦𝜑) | |
6 | 5 | moimi 2508 | . . . . . 6 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥𝜑) |
7 | df-mo 2463 | . . . . . 6 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
8 | 6, 7 | sylib 207 | . . . . 5 ⊢ (∃*𝑥∃𝑦𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑)) |
9 | 4, 8 | eximd 2072 | . . . 4 ⊢ (∃*𝑥∃𝑦𝜑 → (∃𝑦∃𝑥𝜑 → ∃𝑦∃!𝑥𝜑)) |
10 | 2, 9 | syl5bi 231 | . . 3 ⊢ (∃*𝑥∃𝑦𝜑 → (∃𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑)) |
11 | 10 | impcom 445 | . 2 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) → ∃𝑦∃!𝑥𝜑) |
12 | 1, 11 | sylbi 206 | 1 ⊢ (∃!𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∃wex 1695 ∃!weu 2458 ∃*wmo 2459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-eu 2462 df-mo 2463 |
This theorem is referenced by: 2exeu 2537 |
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