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Theorem wl-mo2t 33805
Description: Closed form of mof 2571. (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 df-mo 2565 . 2 (∃*𝑥𝜑 ↔ ∃𝑢𝑥(𝜑𝑥 = 𝑢))
2 nfnf1 2196 . . . 4 𝑦𝑦𝜑
32nfal 2326 . . 3 𝑦𝑥𝑦𝜑
4 nfa1 2193 . . . 4 𝑥𝑥𝑦𝜑
5 sp 2215 . . . . 5 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝜑)
6 nfvd 2010 . . . . 5 (∀𝑥𝑦𝜑 → Ⅎ𝑦 𝑥 = 𝑢)
75, 6nfimd 1992 . . . 4 (∀𝑥𝑦𝜑 → Ⅎ𝑦(𝜑𝑥 = 𝑢))
84, 7nfald 2331 . . 3 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥(𝜑𝑥 = 𝑢))
9 equequ2 2123 . . . . . 6 (𝑢 = 𝑦 → (𝑥 = 𝑢𝑥 = 𝑦))
109imbi2d 331 . . . . 5 (𝑢 = 𝑦 → ((𝜑𝑥 = 𝑢) ↔ (𝜑𝑥 = 𝑦)))
1110albidv 2015 . . . 4 (𝑢 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑢) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
1211a1i 11 . . 3 (∀𝑥𝑦𝜑 → (𝑢 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑢) ↔ ∀𝑥(𝜑𝑥 = 𝑦))))
133, 8, 12cbvexd 2381 . 2 (∀𝑥𝑦𝜑 → (∃𝑢𝑥(𝜑𝑥 = 𝑢) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
141, 13syl5bb 274 1 (∀𝑥𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1650  wex 1874  wnf 1878  ∃*wmo 2563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-ex 1875  df-nf 1879  df-mo 2565
This theorem is referenced by:  wl-mo3t  33806
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