Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 2exeu | Structured version Visualization version GIF version |
Description: Double existential uniqueness implies double unique existential quantification. The converse does not hold. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) |
Ref | Expression |
---|---|
2exeu | ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eumo 2656 | . . . 4 ⊢ (∃!𝑥∃𝑦𝜑 → ∃*𝑥∃𝑦𝜑) | |
2 | euex 2655 | . . . . 5 ⊢ (∃!𝑦𝜑 → ∃𝑦𝜑) | |
3 | 2 | moimi 2620 | . . . 4 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥∃!𝑦𝜑) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (∃!𝑥∃𝑦𝜑 → ∃*𝑥∃!𝑦𝜑) |
5 | 2euex 2719 | . . 3 ⊢ (∃!𝑦∃𝑥𝜑 → ∃𝑥∃!𝑦𝜑) | |
6 | 4, 5 | anim12ci 613 | . 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → (∃𝑥∃!𝑦𝜑 ∧ ∃*𝑥∃!𝑦𝜑)) |
7 | df-eu 2647 | . 2 ⊢ (∃!𝑥∃!𝑦𝜑 ↔ (∃𝑥∃!𝑦𝜑 ∧ ∃*𝑥∃!𝑦𝜑)) | |
8 | 6, 7 | sylibr 235 | 1 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∃wex 1771 ∃*wmo 2613 ∃!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: 2eu1 2728 2eu1OLD 2729 2eu2 2731 2eu3 2732 2eu3OLD 2733 |
Copyright terms: Public domain | W3C validator |