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Theorem wl-eudf 32983
Description: Version of df-eu 2473 with a context and a distinctor replacing a distinct variable condition. This version should be used only to eliminate dv conditions. (Contributed by Wolf Lammen, 23-Sep-2020.)
Hypotheses
Ref Expression
wl-eudf.1 𝑥𝜑
wl-eudf.2 𝑦𝜑
wl-eudf.3 (𝜑 → ¬ ∀𝑥 𝑥 = 𝑦)
wl-eudf.4 (𝜑 → Ⅎ𝑦𝜓)
Assertion
Ref Expression
wl-eudf (𝜑 → (∃!𝑥𝜓 ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦)))

Proof of Theorem wl-eudf
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 df-eu 2473 . 2 (∃!𝑥𝜓 ↔ ∃𝑢𝑥(𝜓𝑥 = 𝑢))
2 wl-eudf.2 . . 3 𝑦𝜑
3 wl-eudf.1 . . . 4 𝑥𝜑
4 wl-eudf.4 . . . . 5 (𝜑 → Ⅎ𝑦𝜓)
5 wl-eudf.3 . . . . . 6 (𝜑 → ¬ ∀𝑥 𝑥 = 𝑦)
6 nfeqf1 2298 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥 = 𝑢)
76naecoms 2312 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 = 𝑢)
85, 7syl 17 . . . . 5 (𝜑 → Ⅎ𝑦 𝑥 = 𝑢)
94, 8nfbid 1829 . . . 4 (𝜑 → Ⅎ𝑦(𝜓𝑥 = 𝑢))
103, 9nfald 2162 . . 3 (𝜑 → Ⅎ𝑦𝑥(𝜓𝑥 = 𝑢))
11 nfnae 2317 . . . . . . 7 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
12 nfeqf2 2296 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑢 = 𝑦)
1311, 12nfan1 2066 . . . . . 6 𝑥(¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑦)
14 equequ2 1950 . . . . . . . 8 (𝑢 = 𝑦 → (𝑥 = 𝑢𝑥 = 𝑦))
1514bibi2d 332 . . . . . . 7 (𝑢 = 𝑦 → ((𝜓𝑥 = 𝑢) ↔ (𝜓𝑥 = 𝑦)))
1615adantl 482 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑦) → ((𝜓𝑥 = 𝑢) ↔ (𝜓𝑥 = 𝑦)))
1713, 16albid 2088 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑦) → (∀𝑥(𝜓𝑥 = 𝑢) ↔ ∀𝑥(𝜓𝑥 = 𝑦)))
185, 17sylan 488 . . . 4 ((𝜑𝑢 = 𝑦) → (∀𝑥(𝜓𝑥 = 𝑢) ↔ ∀𝑥(𝜓𝑥 = 𝑦)))
1918ex 450 . . 3 (𝜑 → (𝑢 = 𝑦 → (∀𝑥(𝜓𝑥 = 𝑢) ↔ ∀𝑥(𝜓𝑥 = 𝑦))))
202, 10, 19cbvexd 2277 . 2 (𝜑 → (∃𝑢𝑥(𝜓𝑥 = 𝑢) ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦)))
211, 20syl5bb 272 1 (𝜑 → (∃!𝑥𝜓 ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1478  wex 1701  wnf 1705  ∃!weu 2469
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 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-eu 2473
This theorem is referenced by:  wl-eutf  32984
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