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Mirrors > Home > MPE Home > Th. List > 2eu2 | Structured version Visualization version GIF version |
Description: Double existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by NM, 3-Dec-2001.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2eu2 | ⊢ (∃!𝑦∃𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ ∃!𝑥∃𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eumo 2662 | . . 3 ⊢ (∃!𝑦∃𝑥𝜑 → ∃*𝑦∃𝑥𝜑) | |
2 | 2moex 2724 | . . 3 ⊢ (∃*𝑦∃𝑥𝜑 → ∀𝑥∃*𝑦𝜑) | |
3 | 2eu1 2734 | . . . 4 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑))) | |
4 | simpl 485 | . . . 4 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃𝑦𝜑) | |
5 | 3, 4 | syl6bi 255 | . . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥∃𝑦𝜑)) |
6 | 1, 2, 5 | 3syl 18 | . 2 ⊢ (∃!𝑦∃𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥∃𝑦𝜑)) |
7 | 2exeu 2730 | . . 3 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑) | |
8 | 7 | expcom 416 | . 2 ⊢ (∃!𝑦∃𝑥𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑥∃!𝑦𝜑)) |
9 | 6, 8 | impbid 214 | 1 ⊢ (∃!𝑦∃𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ ∃!𝑥∃𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1534 ∃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 ax-13 2389 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 |
This theorem is referenced by: 2eu8 2745 |
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