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Theorem reu8 3730
Description: Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)
Hypothesis
Ref Expression
rmo4.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
reu8 (∃!𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem reu8
StepHypRef Expression
1 rmo4.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
21cbvreuvw 3401 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
3 reu6 3723 . 2 (∃!𝑦𝐴 𝜓 ↔ ∃𝑥𝐴𝑦𝐴 (𝜓𝑦 = 𝑥))
4 dfbi2 476 . . . . 5 ((𝜓𝑦 = 𝑥) ↔ ((𝜓𝑦 = 𝑥) ∧ (𝑦 = 𝑥𝜓)))
54ralbii 3094 . . . 4 (∀𝑦𝐴 (𝜓𝑦 = 𝑥) ↔ ∀𝑦𝐴 ((𝜓𝑦 = 𝑥) ∧ (𝑦 = 𝑥𝜓)))
6 r19.26 3112 . . . . 5 (∀𝑦𝐴 ((𝜓𝑦 = 𝑥) ∧ (𝑦 = 𝑥𝜓)) ↔ (∀𝑦𝐴 (𝜓𝑦 = 𝑥) ∧ ∀𝑦𝐴 (𝑦 = 𝑥𝜓)))
7 ancom 462 . . . . . 6 ((𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)) ↔ (∀𝑦𝐴 (𝜓𝑥 = 𝑦) ∧ 𝜑))
8 equcom 2022 . . . . . . . . . 10 (𝑥 = 𝑦𝑦 = 𝑥)
98imbi2i 336 . . . . . . . . 9 ((𝜓𝑥 = 𝑦) ↔ (𝜓𝑦 = 𝑥))
109ralbii 3094 . . . . . . . 8 (∀𝑦𝐴 (𝜓𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝜓𝑦 = 𝑥))
1110a1i 11 . . . . . . 7 (𝑥𝐴 → (∀𝑦𝐴 (𝜓𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝜓𝑦 = 𝑥)))
12 biimt 361 . . . . . . . 8 (𝑥𝐴 → (𝜑 ↔ (𝑥𝐴𝜑)))
13 df-ral 3063 . . . . . . . . 9 (∀𝑦𝐴 (𝑦 = 𝑥𝜓) ↔ ∀𝑦(𝑦𝐴 → (𝑦 = 𝑥𝜓)))
14 bi2.04 389 . . . . . . . . . 10 ((𝑦𝐴 → (𝑦 = 𝑥𝜓)) ↔ (𝑦 = 𝑥 → (𝑦𝐴𝜓)))
1514albii 1822 . . . . . . . . 9 (∀𝑦(𝑦𝐴 → (𝑦 = 𝑥𝜓)) ↔ ∀𝑦(𝑦 = 𝑥 → (𝑦𝐴𝜓)))
16 eleq1w 2817 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1716, 1imbi12d 345 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜓)))
1817bicomd 222 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑦𝐴𝜓) ↔ (𝑥𝐴𝜑)))
1918equcoms 2024 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝑦𝐴𝜓) ↔ (𝑥𝐴𝜑)))
2019equsalvw 2008 . . . . . . . . 9 (∀𝑦(𝑦 = 𝑥 → (𝑦𝐴𝜓)) ↔ (𝑥𝐴𝜑))
2113, 15, 203bitrri 298 . . . . . . . 8 ((𝑥𝐴𝜑) ↔ ∀𝑦𝐴 (𝑦 = 𝑥𝜓))
2212, 21bitrdi 287 . . . . . . 7 (𝑥𝐴 → (𝜑 ↔ ∀𝑦𝐴 (𝑦 = 𝑥𝜓)))
2311, 22anbi12d 632 . . . . . 6 (𝑥𝐴 → ((∀𝑦𝐴 (𝜓𝑥 = 𝑦) ∧ 𝜑) ↔ (∀𝑦𝐴 (𝜓𝑦 = 𝑥) ∧ ∀𝑦𝐴 (𝑦 = 𝑥𝜓))))
247, 23bitrid 283 . . . . 5 (𝑥𝐴 → ((𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)) ↔ (∀𝑦𝐴 (𝜓𝑦 = 𝑥) ∧ ∀𝑦𝐴 (𝑦 = 𝑥𝜓))))
256, 24bitr4id 290 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 ((𝜓𝑦 = 𝑥) ∧ (𝑦 = 𝑥𝜓)) ↔ (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦))))
265, 25bitrid 283 . . 3 (𝑥𝐴 → (∀𝑦𝐴 (𝜓𝑦 = 𝑥) ↔ (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦))))
2726rexbiia 3093 . 2 (∃𝑥𝐴𝑦𝐴 (𝜓𝑦 = 𝑥) ↔ ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
282, 3, 273bitri 297 1 (∃!𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540  wcel 2107  wral 3062  wrex 3071  ∃!wreu 3375
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-10 2138  ax-12 2172
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clel 2811  df-ral 3063  df-rex 3072  df-reu 3378
This theorem is referenced by:  reu8nf  3872  updjud  9929  reusq0  15409  reumodprminv  16737  grpinveu  18859  addsq2reu  26943  2sqreulem1  26949  2sqreunnlem1  26952  grpoideu  29762  grpoinveu  29772  cvmlift3lem2  34311  euoreqb  45817  2reu8i  45821  2reuimp0  45822  paireqne  46179  itsclquadeu  47463
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