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Theorem wl-dfrmov 35012
Description: Alternate definition of restricted "at most one" (df-wl-rmo 35010) when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 28-May-2023.)
Assertion
Ref Expression
wl-dfrmov (∃*(𝑥 : 𝐴)𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem wl-dfrmov
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 wl-dfralv 34999 . . . 4 (∀(𝑥 : 𝐴)(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
2 impexp 454 . . . . 5 (((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → (𝜑𝑥 = 𝑦)))
32albii 1821 . . . 4 (∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
41, 3bitr4i 281 . . 3 (∀(𝑥 : 𝐴)(𝜑𝑥 = 𝑦) ↔ ∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦))
54exbii 1849 . 2 (∃𝑦∀(𝑥 : 𝐴)(𝜑𝑥 = 𝑦) ↔ ∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦))
6 df-wl-rmo 35010 . 2 (∃*(𝑥 : 𝐴)𝜑 ↔ ∃𝑦∀(𝑥 : 𝐴)(𝜑𝑥 = 𝑦))
7 df-mo 2601 . 2 (∃*𝑥(𝑥𝐴𝜑) ↔ ∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦))
85, 6, 73bitr4i 306 1 (∃*(𝑥 : 𝐴)𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536  wex 1781  wcel 2112  ∃*wmo 2599  wl-ral 34989  ∃*wl-rmo 34991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-11 2159
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-mo 2601  df-clel 2873  df-wl-ral 34994  df-wl-rmo 35010
This theorem is referenced by:  wl-dfreuv  35016
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