Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 2euswapv | Structured version Visualization version GIF version |
Description: A condition allowing to swap an existential quantifier and a unique existential quantifier. Version of 2euswap 2729 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 10-Apr-2004.) (Revised by Gino Giotto, 22-Aug-2023.) |
Ref | Expression |
---|---|
2euswapv | ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | excomim 2169 | . . . 4 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑) | |
2 | 1 | a1i 11 | . . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)) |
3 | 2moswapv 2713 | . . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑)) | |
4 | 2, 3 | anim12d 610 | . 2 ⊢ (∀𝑥∃*𝑦𝜑 → ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) → (∃𝑦∃𝑥𝜑 ∧ ∃*𝑦∃𝑥𝜑))) |
5 | df-eu 2653 | . 2 ⊢ (∃!𝑥∃𝑦𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑)) | |
6 | df-eu 2653 | . 2 ⊢ (∃!𝑦∃𝑥𝜑 ↔ (∃𝑦∃𝑥𝜑 ∧ ∃*𝑦∃𝑥𝜑)) | |
7 | 4, 5, 6 | 3imtr4g 298 | 1 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ 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 |
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: 2eu1v 2735 euxfr2w 3707 2reuswap 3733 2reuswap2 3734 reuxfrdf 30251 |
Copyright terms: Public domain | W3C validator |