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Theorem moexex 2570
Description: "At most one" double quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.)
Hypothesis
Ref Expression
moexex.1 𝑦𝜑
Assertion
Ref Expression
moexex ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦𝑥(𝜑𝜓))

Proof of Theorem moexex
StepHypRef Expression
1 nfmo1 2509 . . . 4 𝑥∃*𝑥𝜑
2 nfa1 2068 . . . . 5 𝑥𝑥∃*𝑦𝜓
3 nfe1 2067 . . . . . 6 𝑥𝑥(𝜑𝜓)
43nfmo 2515 . . . . 5 𝑥∃*𝑦𝑥(𝜑𝜓)
52, 4nfim 1865 . . . 4 𝑥(∀𝑥∃*𝑦𝜓 → ∃*𝑦𝑥(𝜑𝜓))
6 moexex.1 . . . . . . 7 𝑦𝜑
76nfmo 2515 . . . . . 6 𝑦∃*𝑥𝜑
8 mopick 2564 . . . . . . . 8 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
98ex 449 . . . . . . 7 (∃*𝑥𝜑 → (∃𝑥(𝜑𝜓) → (𝜑𝜓)))
109com23 86 . . . . . 6 (∃*𝑥𝜑 → (𝜑 → (∃𝑥(𝜑𝜓) → 𝜓)))
117, 6, 10alrimd 2122 . . . . 5 (∃*𝑥𝜑 → (𝜑 → ∀𝑦(∃𝑥(𝜑𝜓) → 𝜓)))
12 moim 2548 . . . . . 6 (∀𝑦(∃𝑥(𝜑𝜓) → 𝜓) → (∃*𝑦𝜓 → ∃*𝑦𝑥(𝜑𝜓)))
1312spsd 2095 . . . . 5 (∀𝑦(∃𝑥(𝜑𝜓) → 𝜓) → (∀𝑥∃*𝑦𝜓 → ∃*𝑦𝑥(𝜑𝜓)))
1411, 13syl6 35 . . . 4 (∃*𝑥𝜑 → (𝜑 → (∀𝑥∃*𝑦𝜓 → ∃*𝑦𝑥(𝜑𝜓))))
151, 5, 14exlimd 2125 . . 3 (∃*𝑥𝜑 → (∃𝑥𝜑 → (∀𝑥∃*𝑦𝜓 → ∃*𝑦𝑥(𝜑𝜓))))
166nfex 2192 . . . . . . 7 𝑦𝑥𝜑
17 exsimpl 1835 . . . . . . 7 (∃𝑥(𝜑𝜓) → ∃𝑥𝜑)
1816, 17exlimi 2124 . . . . . 6 (∃𝑦𝑥(𝜑𝜓) → ∃𝑥𝜑)
19 exmo 2523 . . . . . . 7 (∃𝑦𝑥(𝜑𝜓) ∨ ∃*𝑦𝑥(𝜑𝜓))
2019ori 389 . . . . . 6 (¬ ∃𝑦𝑥(𝜑𝜓) → ∃*𝑦𝑥(𝜑𝜓))
2118, 20nsyl4 156 . . . . 5 (¬ ∃*𝑦𝑥(𝜑𝜓) → ∃𝑥𝜑)
2221con1i 144 . . . 4 (¬ ∃𝑥𝜑 → ∃*𝑦𝑥(𝜑𝜓))
2322a1d 25 . . 3 (¬ ∃𝑥𝜑 → (∀𝑥∃*𝑦𝜓 → ∃*𝑦𝑥(𝜑𝜓)))
2415, 23pm2.61d1 171 . 2 (∃*𝑥𝜑 → (∀𝑥∃*𝑦𝜓 → ∃*𝑦𝑥(𝜑𝜓)))
2524imp 444 1 ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wal 1521  wex 1744  wnf 1748  ∃*wmo 2499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-eu 2502  df-mo 2503
This theorem is referenced by:  moexexv  2571  2moswap  2576
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