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Theorem reu8 3435
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))
21cbvreuv 3203 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
3 reu6 3428 . 2 (∃!𝑦𝐴 𝜓 ↔ ∃𝑥𝐴𝑦𝐴 (𝜓𝑦 = 𝑥))
4 dfbi2 661 . . . . 5 ((𝜓𝑦 = 𝑥) ↔ ((𝜓𝑦 = 𝑥) ∧ (𝑦 = 𝑥𝜓)))
54ralbii 3009 . . . 4 (∀𝑦𝐴 (𝜓𝑦 = 𝑥) ↔ ∀𝑦𝐴 ((𝜓𝑦 = 𝑥) ∧ (𝑦 = 𝑥𝜓)))
6 ancom 465 . . . . . 6 ((𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)) ↔ (∀𝑦𝐴 (𝜓𝑥 = 𝑦) ∧ 𝜑))
7 equcom 1991 . . . . . . . . . 10 (𝑥 = 𝑦𝑦 = 𝑥)
87imbi2i 325 . . . . . . . . 9 ((𝜓𝑥 = 𝑦) ↔ (𝜓𝑦 = 𝑥))
98ralbii 3009 . . . . . . . 8 (∀𝑦𝐴 (𝜓𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝜓𝑦 = 𝑥))
109a1i 11 . . . . . . 7 (𝑥𝐴 → (∀𝑦𝐴 (𝜓𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝜓𝑦 = 𝑥)))
11 biimt 349 . . . . . . . 8 (𝑥𝐴 → (𝜑 ↔ (𝑥𝐴𝜑)))
12 df-ral 2946 . . . . . . . . 9 (∀𝑦𝐴 (𝑦 = 𝑥𝜓) ↔ ∀𝑦(𝑦𝐴 → (𝑦 = 𝑥𝜓)))
13 bi2.04 375 . . . . . . . . . 10 ((𝑦𝐴 → (𝑦 = 𝑥𝜓)) ↔ (𝑦 = 𝑥 → (𝑦𝐴𝜓)))
1413albii 1787 . . . . . . . . 9 (∀𝑦(𝑦𝐴 → (𝑦 = 𝑥𝜓)) ↔ ∀𝑦(𝑦 = 𝑥 → (𝑦𝐴𝜓)))
15 eleq1w 2713 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1615, 1imbi12d 333 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜓)))
1716bicomd 213 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑦𝐴𝜓) ↔ (𝑥𝐴𝜑)))
1817equcoms 1993 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝑦𝐴𝜓) ↔ (𝑥𝐴𝜑)))
1918equsalvw 1977 . . . . . . . . 9 (∀𝑦(𝑦 = 𝑥 → (𝑦𝐴𝜓)) ↔ (𝑥𝐴𝜑))
2012, 14, 193bitrri 287 . . . . . . . 8 ((𝑥𝐴𝜑) ↔ ∀𝑦𝐴 (𝑦 = 𝑥𝜓))
2111, 20syl6bb 276 . . . . . . 7 (𝑥𝐴 → (𝜑 ↔ ∀𝑦𝐴 (𝑦 = 𝑥𝜓)))
2210, 21anbi12d 747 . . . . . 6 (𝑥𝐴 → ((∀𝑦𝐴 (𝜓𝑥 = 𝑦) ∧ 𝜑) ↔ (∀𝑦𝐴 (𝜓𝑦 = 𝑥) ∧ ∀𝑦𝐴 (𝑦 = 𝑥𝜓))))
236, 22syl5bb 272 . . . . 5 (𝑥𝐴 → ((𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)) ↔ (∀𝑦𝐴 (𝜓𝑦 = 𝑥) ∧ ∀𝑦𝐴 (𝑦 = 𝑥𝜓))))
24 r19.26 3093 . . . . 5 (∀𝑦𝐴 ((𝜓𝑦 = 𝑥) ∧ (𝑦 = 𝑥𝜓)) ↔ (∀𝑦𝐴 (𝜓𝑦 = 𝑥) ∧ ∀𝑦𝐴 (𝑦 = 𝑥𝜓)))
2523, 24syl6rbbr 279 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 ((𝜓𝑦 = 𝑥) ∧ (𝑦 = 𝑥𝜓)) ↔ (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦))))
265, 25syl5bb 272 . . 3 (𝑥𝐴 → (∀𝑦𝐴 (𝜓𝑦 = 𝑥) ↔ (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦))))
2726rexbiia 3069 . 2 (∃𝑥𝐴𝑦𝐴 (𝜓𝑦 = 𝑥) ↔ ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
282, 3, 273bitri 286 1 (∃!𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521  wcel 2030  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-ral 2946  df-rex 2947  df-reu 2948
This theorem is referenced by:  reu8nf  3549  reumodprminv  15556  grpinveu  17503  grpoideu  27491  grpoinveu  27501  cvmlift3lem2  31428  reuccatpfxs1  41759
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