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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. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker 2exeuv 2634 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2exeu | ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eumo 2578 | . . . 4 ⊢ (∃!𝑥∃𝑦𝜑 → ∃*𝑥∃𝑦𝜑) | |
2 | euex 2577 | . . . . 5 ⊢ (∃!𝑦𝜑 → ∃𝑦𝜑) | |
3 | 2 | moimi 2545 | . . . 4 ⊢ (∃*𝑥∃𝑦𝜑 → ∃*𝑥∃!𝑦𝜑) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (∃!𝑥∃𝑦𝜑 → ∃*𝑥∃!𝑦𝜑) |
5 | 2euex 2643 | . . 3 ⊢ (∃!𝑦∃𝑥𝜑 → ∃𝑥∃!𝑦𝜑) | |
6 | 4, 5 | anim12ci 614 | . 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → (∃𝑥∃!𝑦𝜑 ∧ ∃*𝑥∃!𝑦𝜑)) |
7 | df-eu 2569 | . 2 ⊢ (∃!𝑥∃!𝑦𝜑 ↔ (∃𝑥∃!𝑦𝜑 ∧ ∃*𝑥∃!𝑦𝜑)) | |
8 | 6, 7 | sylibr 233 | 1 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∃wex 1782 ∃*wmo 2538 ∃!weu 2568 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-11 2154 ax-12 2171 ax-13 2372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-nf 1787 df-mo 2540 df-eu 2569 |
This theorem is referenced by: 2eu1 2652 2eu2 2654 2eu3 2655 |
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