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Mirrors > Home > MPE Home > Th. List > moexex | Structured version Visualization version GIF version |
Description: "At most one" double quantification. Usage of this theorem is discouraged because it depends on ax-13 2374. Use the version moexexvw 2625 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.) Factor out common proof lines with moexexvw 2625. (Revised by Wolf Lammen, 2-Oct-2023.) (New usage is discouraged.) |
Ref | Expression |
---|---|
moexex.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
moexex | ⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moexex.1 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfmo 2559 | . 2 ⊢ Ⅎ𝑦∃*𝑥𝜑 |
3 | nfe1 2147 | . . 3 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝜓) | |
4 | 3 | nfmo 2559 | . 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 Ⅎwnf 1779 ∃*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 ax-13 2374 |
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: moexexv 2636 2moswap 2641 |
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