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Mirrors > Home > MPE Home > Th. List > 2euswap | Structured version Visualization version GIF version |
Description: A condition allowing to swap an existential quantifier and a unique existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker 2euswapv 2633 when possible. (Contributed by NM, 10-Apr-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2euswap | ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | excomim 2170 | . . . 4 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑) | |
2 | 1 | a1i 11 | . . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)) |
3 | 2moswap 2647 | . . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑)) | |
4 | 2, 3 | anim12d 612 | . 2 ⊢ (∀𝑥∃*𝑦𝜑 → ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) → (∃𝑦∃𝑥𝜑 ∧ ∃*𝑦∃𝑥𝜑))) |
5 | df-eu 2570 | . 2 ⊢ (∃!𝑥∃𝑦𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑)) | |
6 | df-eu 2570 | . 2 ⊢ (∃!𝑦∃𝑥𝜑 ↔ (∃𝑦∃𝑥𝜑 ∧ ∃*𝑦∃𝑥𝜑)) | |
7 | 4, 5, 6 | 3imtr4g 299 | 1 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1540 ∃wex 1786 ∃*wmo 2538 ∃!weu 2569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-10 2144 ax-11 2161 ax-12 2178 ax-13 2371 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 df-mo 2540 df-eu 2570 |
This theorem is referenced by: 2eu1 2653 euxfr2 3619 |
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