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Mirrors > Home > MPE Home > Th. List > 2euswap | Structured version Visualization version GIF version |
Description: A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 10-Apr-2004.) |
Ref | Expression |
---|---|
2euswap | ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | excomim 2192 | . . . 4 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑) | |
2 | 1 | a1i 11 | . . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)) |
3 | 2moswap 2685 | . . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑)) | |
4 | 2, 3 | anim12d 587 | . 2 ⊢ (∀𝑥∃*𝑦𝜑 → ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) → (∃𝑦∃𝑥𝜑 ∧ ∃*𝑦∃𝑥𝜑))) |
5 | eu5 2633 | . 2 ⊢ (∃!𝑥∃𝑦𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑)) | |
6 | eu5 2633 | . 2 ⊢ (∃!𝑦∃𝑥𝜑 ↔ (∃𝑦∃𝑥𝜑 ∧ ∃*𝑦∃𝑥𝜑)) | |
7 | 4, 5, 6 | 3imtr4g 285 | 1 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1630 ∃wex 1853 ∃!weu 2607 ∃*wmo 2608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-eu 2611 df-mo 2612 |
This theorem is referenced by: 2eu1 2691 euxfr2 3532 2reuswap 3551 2reuswap2 29657 |
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