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Theorem reu8nf 3784
 Description: Restricted uniqueness using implicit substitution. This version of reu8 3648 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 1916 . . 3 𝑤𝜑
2 reu8nf.2 . . 3 𝑥𝜒
3 reu8nf.3 . . 3 (𝑥 = 𝑤 → (𝜑𝜒))
41, 2, 3cbvreuw 3355 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑤𝐴 𝜒)
5 reu8nf.4 . . 3 (𝑤 = 𝑦 → (𝜒𝜓))
65reu8 3648 . 2 (∃!𝑤𝐴 𝜒 ↔ ∃𝑤𝐴 (𝜒 ∧ ∀𝑦𝐴 (𝜓𝑤 = 𝑦)))
7 nfcv 2920 . . . . 5 𝑥𝐴
8 reu8nf.1 . . . . . 6 𝑥𝜓
9 nfv 1916 . . . . . 6 𝑥 𝑤 = 𝑦
108, 9nfim 1898 . . . . 5 𝑥(𝜓𝑤 = 𝑦)
117, 10nfralw 3154 . . . 4 𝑥𝑦𝐴 (𝜓𝑤 = 𝑦)
122, 11nfan 1901 . . 3 𝑥(𝜒 ∧ ∀𝑦𝐴 (𝜓𝑤 = 𝑦))
13 nfv 1916 . . 3 𝑤(𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦))
143bicomd 226 . . . . 5 (𝑥 = 𝑤 → (𝜒𝜑))
1514equcoms 2028 . . . 4 (𝑤 = 𝑥 → (𝜒𝜑))
16 equequ1 2033 . . . . . 6 (𝑤 = 𝑥 → (𝑤 = 𝑦𝑥 = 𝑦))
1716imbi2d 345 . . . . 5 (𝑤 = 𝑥 → ((𝜓𝑤 = 𝑦) ↔ (𝜓𝑥 = 𝑦)))
1817ralbidv 3127 . . . 4 (𝑤 = 𝑥 → (∀𝑦𝐴 (𝜓𝑤 = 𝑦) ↔ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
1915, 18anbi12d 634 . . 3 (𝑤 = 𝑥 → ((𝜒 ∧ ∀𝑦𝐴 (𝜓𝑤 = 𝑦)) ↔ (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦))))
2012, 13, 19cbvrexw 3354 . 2 (∃𝑤𝐴 (𝜒 ∧ ∀𝑦𝐴 (𝜓𝑤 = 𝑦)) ↔ ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
214, 6, 203bitri 301 1 (∃!𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  Ⅎwnf 1786  ∀wral 3071  ∃wrex 3072  ∃!wreu 3073 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-reu 3078 This theorem is referenced by:  reusngf  4570  reuprg0  4596  reuop  6123  reuccatpfxs1  14157  reupr  44408
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