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Theorem reu8nf 3549
Description: Restricted uniqueness using implicit substitution. This version of reu8 3435 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 1883 . . 3 𝑤𝜑
2 reu8nf.2 . . 3 𝑥𝜒
3 reu8nf.3 . . 3 (𝑥 = 𝑤 → (𝜑𝜒))
41, 2, 3cbvreu 3199 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑤𝐴 𝜒)
5 reu8nf.4 . . 3 (𝑤 = 𝑦 → (𝜒𝜓))
65reu8 3435 . 2 (∃!𝑤𝐴 𝜒 ↔ ∃𝑤𝐴 (𝜒 ∧ ∀𝑦𝐴 (𝜓𝑤 = 𝑦)))
7 nfcv 2793 . . . . 5 𝑥𝐴
8 reu8nf.1 . . . . . 6 𝑥𝜓
9 nfv 1883 . . . . . 6 𝑥 𝑤 = 𝑦
108, 9nfim 1865 . . . . 5 𝑥(𝜓𝑤 = 𝑦)
117, 10nfral 2974 . . . 4 𝑥𝑦𝐴 (𝜓𝑤 = 𝑦)
122, 11nfan 1868 . . 3 𝑥(𝜒 ∧ ∀𝑦𝐴 (𝜓𝑤 = 𝑦))
13 nfv 1883 . . 3 𝑤(𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦))
143bicomd 213 . . . . 5 (𝑥 = 𝑤 → (𝜒𝜑))
1514equcoms 1993 . . . 4 (𝑤 = 𝑥 → (𝜒𝜑))
16 equequ1 1998 . . . . . 6 (𝑤 = 𝑥 → (𝑤 = 𝑦𝑥 = 𝑦))
1716imbi2d 329 . . . . 5 (𝑤 = 𝑥 → ((𝜓𝑤 = 𝑦) ↔ (𝜓𝑥 = 𝑦)))
1817ralbidv 3015 . . . 4 (𝑤 = 𝑥 → (∀𝑦𝐴 (𝜓𝑤 = 𝑦) ↔ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
1915, 18anbi12d 747 . . 3 (𝑤 = 𝑥 → ((𝜒 ∧ ∀𝑦𝐴 (𝜓𝑤 = 𝑦)) ↔ (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦))))
2012, 13, 19cbvrex 3198 . 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 383  wnf 1748  wral 2941  wrex 2942  ∃!wreu 2943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-reu 2948
This theorem is referenced by:  reuccats1  13526
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