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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 2723 with an additional disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 26-Jan-1997.) (Revised by Gino Giotto, 22-Aug-2023.) Factor out common proof lines with moexex 2722. (Revised by Wolf Lammen, 2-Oct-2023.) |
Ref | Expression |
---|---|
moexexvw | ⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1914 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1914 | . 2 ⊢ Ⅎ𝑦∃*𝑥𝜑 | |
3 | nfe1 2153 | . . 3 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝜓) | |
4 | 3 | nfmov 2643 | . 2 ⊢ Ⅎ𝑥∃*𝑦∃𝑥(𝜑 ∧ 𝜓) |
5 | 1, 2, 4 | moexexlem 2710 | 1 ⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1534 ∃wex 1779 ∃*wmo 2619 |
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 |
This theorem is referenced by: mosub 3700 funco 6388 |
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