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Theorem reu8 3384
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 3161 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
3 reu6 3377 . 2 (∃!𝑦𝐴 𝜓 ↔ ∃𝑥𝐴𝑦𝐴 (𝜓𝑦 = 𝑥))
4 dfbi2 659 . . . . 5 ((𝜓𝑦 = 𝑥) ↔ ((𝜓𝑦 = 𝑥) ∧ (𝑦 = 𝑥𝜓)))
54ralbii 2974 . . . 4 (∀𝑦𝐴 (𝜓𝑦 = 𝑥) ↔ ∀𝑦𝐴 ((𝜓𝑦 = 𝑥) ∧ (𝑦 = 𝑥𝜓)))
6 ancom 466 . . . . . 6 ((𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)) ↔ (∀𝑦𝐴 (𝜓𝑥 = 𝑦) ∧ 𝜑))
7 equcom 1942 . . . . . . . . . 10 (𝑥 = 𝑦𝑦 = 𝑥)
87imbi2i 326 . . . . . . . . 9 ((𝜓𝑥 = 𝑦) ↔ (𝜓𝑦 = 𝑥))
98ralbii 2974 . . . . . . . 8 (∀𝑦𝐴 (𝜓𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝜓𝑦 = 𝑥))
109a1i 11 . . . . . . 7 (𝑥𝐴 → (∀𝑦𝐴 (𝜓𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝜓𝑦 = 𝑥)))
11 biimt 350 . . . . . . . 8 (𝑥𝐴 → (𝜑 ↔ (𝑥𝐴𝜑)))
12 df-ral 2912 . . . . . . . . 9 (∀𝑦𝐴 (𝑦 = 𝑥𝜓) ↔ ∀𝑦(𝑦𝐴 → (𝑦 = 𝑥𝜓)))
13 bi2.04 376 . . . . . . . . . 10 ((𝑦𝐴 → (𝑦 = 𝑥𝜓)) ↔ (𝑦 = 𝑥 → (𝑦𝐴𝜓)))
1413albii 1744 . . . . . . . . 9 (∀𝑦(𝑦𝐴 → (𝑦 = 𝑥𝜓)) ↔ ∀𝑦(𝑦 = 𝑥 → (𝑦𝐴𝜓)))
15 eleq1 2686 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1615, 1imbi12d 334 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜓)))
1716bicomd 213 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑦𝐴𝜓) ↔ (𝑥𝐴𝜑)))
1817equcoms 1944 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝑦𝐴𝜓) ↔ (𝑥𝐴𝜑)))
1918equsalvw 1928 . . . . . . . . 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 746 . . . . . 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 3057 . . . . 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 3033 . 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 384  wal 1478  wcel 1987  wral 2907  wrex 2908  ∃!wreu 2909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-cleq 2614  df-clel 2617  df-ral 2912  df-rex 2913  df-reu 2914
This theorem is referenced by:  reuccats1  13418  reumodprminv  15433  grpinveu  17377  grpoideu  27209  grpoinveu  27219  cvmlift3lem2  31007  reuccatpfxs1  40730
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