MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reu8nf Structured version   Visualization version   GIF version

Theorem reu8nf 3665
Description: Restricted uniqueness using implicit substitution. This version of reu8 3554 uses a non-freeness hypothesis for 𝑥 and 𝜓 instead of distinct variable conditions. (Contributed by AV, 21-Jan-2022.)
Hypotheses
Ref Expression
reu8nf.1 𝑥𝜓
reu8nf.2 𝑥𝜒
reu8nf.3 (𝑥 = 𝑤 → (𝜑𝜒))
reu8nf.4 (𝑤 = 𝑦 → (𝜒𝜓))
Assertion
Ref Expression
reu8nf (∃!𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
Distinct variable groups:   𝑥,𝑤,𝑦,𝐴   𝜑,𝑤   𝜓,𝑤   𝜒,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑤)

Proof of Theorem reu8nf
StepHypRef Expression
1 nfv 1995 . . 3 𝑤𝜑
2 reu8nf.2 . . 3 𝑥𝜒
3 reu8nf.3 . . 3 (𝑥 = 𝑤 → (𝜑𝜒))
41, 2, 3cbvreu 3318 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑤𝐴 𝜒)
5 reu8nf.4 . . 3 (𝑤 = 𝑦 → (𝜒𝜓))
65reu8 3554 . 2 (∃!𝑤𝐴 𝜒 ↔ ∃𝑤𝐴 (𝜒 ∧ ∀𝑦𝐴 (𝜓𝑤 = 𝑦)))
7 nfcv 2913 . . . . 5 𝑥𝐴
8 reu8nf.1 . . . . . 6 𝑥𝜓
9 nfv 1995 . . . . . 6 𝑥 𝑤 = 𝑦
108, 9nfim 1977 . . . . 5 𝑥(𝜓𝑤 = 𝑦)
117, 10nfral 3094 . . . 4 𝑥𝑦𝐴 (𝜓𝑤 = 𝑦)
122, 11nfan 1980 . . 3 𝑥(𝜒 ∧ ∀𝑦𝐴 (𝜓𝑤 = 𝑦))
13 nfv 1995 . . 3 𝑤(𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦))
143bicomd 213 . . . . 5 (𝑥 = 𝑤 → (𝜒𝜑))
1514equcoms 2105 . . . 4 (𝑤 = 𝑥 → (𝜒𝜑))
16 equequ1 2110 . . . . . 6 (𝑤 = 𝑥 → (𝑤 = 𝑦𝑥 = 𝑦))
1716imbi2d 329 . . . . 5 (𝑤 = 𝑥 → ((𝜓𝑤 = 𝑦) ↔ (𝜓𝑥 = 𝑦)))
1817ralbidv 3135 . . . 4 (𝑤 = 𝑥 → (∀𝑦𝐴 (𝜓𝑤 = 𝑦) ↔ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
1915, 18anbi12d 616 . . 3 (𝑤 = 𝑥 → ((𝜒 ∧ ∀𝑦𝐴 (𝜓𝑤 = 𝑦)) ↔ (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦))))
2012, 13, 19cbvrex 3317 . 2 (∃𝑤𝐴 (𝜒 ∧ ∀𝑦𝐴 (𝜓𝑤 = 𝑦)) ↔ ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
214, 6, 203bitri 286 1 (∃!𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wnf 1856  wral 3061  wrex 3062  ∃!wreu 3063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-reu 3068
This theorem is referenced by:  reuccats1  13689
  Copyright terms: Public domain W3C validator