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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 2366. Use the version moexexvw 2617 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.) Factor out common proof lines with moexexvw 2617. (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 2551 | . 2 ⊢ Ⅎ𝑦∃*𝑥𝜑 |
3 | nfe1 2140 | . . 3 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝜓) | |
4 | 3 | nfmo 2551 | . 2 ⊢ Ⅎ𝑥∃*𝑦∃𝑥(𝜑 ∧ 𝜓) |
5 | 1, 2, 4 | moexexlem 2615 | 1 ⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∀wal 1532 ∃wex 1774 Ⅎwnf 1778 ∃*wmo 2527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-10 2130 ax-11 2147 ax-12 2167 ax-13 2366 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1537 df-ex 1775 df-nf 1779 df-mo 2529 |
This theorem is referenced by: moexexv 2628 2moswap 2633 |
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