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Mirrors > Home > MPE Home > Th. List > moexexvw | Structured version Visualization version GIF version |
Description: "At most one" double quantification. Version of moexexv 2636 with an additional disjoint variable condition, which does not require ax-13 2374. (Contributed by NM, 26-Jan-1997.) (Revised by GG, 22-Aug-2023.) Factor out common proof lines with moexex 2635. (Revised by Wolf Lammen, 2-Oct-2023.) |
Ref | Expression |
---|---|
moexexvw | ⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1911 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1911 | . 2 ⊢ Ⅎ𝑦∃*𝑥𝜑 | |
3 | nfe1 2147 | . . 3 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝜓) | |
4 | 3 | nfmov 2557 | . 2 ⊢ Ⅎ𝑥∃*𝑦∃𝑥(𝜑 ∧ 𝜓) |
5 | 1, 2, 4 | moexexlem 2623 | 1 ⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1534 ∃wex 1775 ∃*wmo 2535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-10 2138 ax-11 2154 ax-12 2174 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-ex 1776 df-nf 1780 df-mo 2537 |
This theorem is referenced by: mosub 3721 funco 6607 tfsconcatlem 43325 |
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