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Theorem wl-mo2t 32984
Description: Closed form of mo2 2483. (Contributed by Wolf Lammen, 18-Aug-2019.)
Assertion
Ref Expression
wl-mo2t (∀𝑥𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem wl-mo2t
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 mo2v 2481 . 2 (∃*𝑥𝜑 ↔ ∃𝑢𝑥(𝜑𝑥 = 𝑢))
2 nfnf1 2033 . . . 4 𝑦𝑦𝜑
32nfal 2155 . . 3 𝑦𝑥𝑦𝜑
4 nfa1 2030 . . . 4 𝑥𝑥𝑦𝜑
5 sp 2056 . . . . 5 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝜑)
6 nfvd 1846 . . . . 5 (∀𝑥𝑦𝜑 → Ⅎ𝑦 𝑥 = 𝑢)
75, 6nfimd 1825 . . . 4 (∀𝑥𝑦𝜑 → Ⅎ𝑦(𝜑𝑥 = 𝑢))
84, 7nfald 2167 . . 3 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥(𝜑𝑥 = 𝑢))
9 equequ2 1955 . . . . . 6 (𝑢 = 𝑦 → (𝑥 = 𝑢𝑥 = 𝑦))
109imbi2d 330 . . . . 5 (𝑢 = 𝑦 → ((𝜑𝑥 = 𝑢) ↔ (𝜑𝑥 = 𝑦)))
1110albidv 1851 . . . 4 (𝑢 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑢) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
1211a1i 11 . . 3 (∀𝑥𝑦𝜑 → (𝑢 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑢) ↔ ∀𝑥(𝜑𝑥 = 𝑦))))
133, 8, 12cbvexd 2282 . 2 (∀𝑥𝑦𝜑 → (∃𝑢𝑥(𝜑𝑥 = 𝑢) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
141, 13syl5bb 272 1 (∀𝑥𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478  wex 1701  wnf 1705  ∃*wmo 2475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-eu 2478  df-mo 2479
This theorem is referenced by:  wl-mo3t  32985
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