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Theorem reu8 2948
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 2720 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
3 reu6 2941 . 2 (∃!𝑦𝐴 𝜓 ↔ ∃𝑥𝐴𝑦𝐴 (𝜓𝑦 = 𝑥))
4 dfbi2 388 . . . . 5 ((𝜓𝑦 = 𝑥) ↔ ((𝜓𝑦 = 𝑥) ∧ (𝑦 = 𝑥𝜓)))
54ralbii 2496 . . . 4 (∀𝑦𝐴 (𝜓𝑦 = 𝑥) ↔ ∀𝑦𝐴 ((𝜓𝑦 = 𝑥) ∧ (𝑦 = 𝑥𝜓)))
6 r19.26 2616 . . . . 5 (∀𝑦𝐴 ((𝜓𝑦 = 𝑥) ∧ (𝑦 = 𝑥𝜓)) ↔ (∀𝑦𝐴 (𝜓𝑦 = 𝑥) ∧ ∀𝑦𝐴 (𝑦 = 𝑥𝜓)))
7 ancom 266 . . . . . 6 ((𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)) ↔ (∀𝑦𝐴 (𝜓𝑥 = 𝑦) ∧ 𝜑))
8 equcom 1717 . . . . . . . . . 10 (𝑥 = 𝑦𝑦 = 𝑥)
98imbi2i 226 . . . . . . . . 9 ((𝜓𝑥 = 𝑦) ↔ (𝜓𝑦 = 𝑥))
109ralbii 2496 . . . . . . . 8 (∀𝑦𝐴 (𝜓𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝜓𝑦 = 𝑥))
1110a1i 9 . . . . . . 7 (𝑥𝐴 → (∀𝑦𝐴 (𝜓𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝜓𝑦 = 𝑥)))
12 biimt 241 . . . . . . . 8 (𝑥𝐴 → (𝜑 ↔ (𝑥𝐴𝜑)))
13 df-ral 2473 . . . . . . . . 9 (∀𝑦𝐴 (𝑦 = 𝑥𝜓) ↔ ∀𝑦(𝑦𝐴 → (𝑦 = 𝑥𝜓)))
14 bi2.04 248 . . . . . . . . . 10 ((𝑦𝐴 → (𝑦 = 𝑥𝜓)) ↔ (𝑦 = 𝑥 → (𝑦𝐴𝜓)))
1514albii 1481 . . . . . . . . 9 (∀𝑦(𝑦𝐴 → (𝑦 = 𝑥𝜓)) ↔ ∀𝑦(𝑦 = 𝑥 → (𝑦𝐴𝜓)))
16 vex 2755 . . . . . . . . . 10 𝑥 ∈ V
17 eleq1 2252 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1817, 1imbi12d 234 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜓)))
1918bicomd 141 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑦𝐴𝜓) ↔ (𝑥𝐴𝜑)))
2019equcoms 1719 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝑦𝐴𝜓) ↔ (𝑥𝐴𝜑)))
2116, 20ceqsalv 2782 . . . . . . . . 9 (∀𝑦(𝑦 = 𝑥 → (𝑦𝐴𝜓)) ↔ (𝑥𝐴𝜑))
2213, 15, 213bitrri 207 . . . . . . . 8 ((𝑥𝐴𝜑) ↔ ∀𝑦𝐴 (𝑦 = 𝑥𝜓))
2312, 22bitrdi 196 . . . . . . 7 (𝑥𝐴 → (𝜑 ↔ ∀𝑦𝐴 (𝑦 = 𝑥𝜓)))
2411, 23anbi12d 473 . . . . . 6 (𝑥𝐴 → ((∀𝑦𝐴 (𝜓𝑥 = 𝑦) ∧ 𝜑) ↔ (∀𝑦𝐴 (𝜓𝑦 = 𝑥) ∧ ∀𝑦𝐴 (𝑦 = 𝑥𝜓))))
257, 24bitrid 192 . . . . 5 (𝑥𝐴 → ((𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)) ↔ (∀𝑦𝐴 (𝜓𝑦 = 𝑥) ∧ ∀𝑦𝐴 (𝑦 = 𝑥𝜓))))
266, 25bitr4id 199 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 ((𝜓𝑦 = 𝑥) ∧ (𝑦 = 𝑥𝜓)) ↔ (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦))))
275, 26bitrid 192 . . 3 (𝑥𝐴 → (∀𝑦𝐴 (𝜓𝑦 = 𝑥) ↔ (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦))))
2827rexbiia 2505 . 2 (∃𝑥𝐴𝑦𝐴 (𝜓𝑦 = 𝑥) ↔ ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
292, 3, 283bitri 206 1 (∃!𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓𝑥 = 𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1362  wcel 2160  wral 2468  wrex 2469  ∃!wreu 2470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-clab 2176  df-cleq 2182  df-clel 2185  df-ral 2473  df-rex 2474  df-reu 2475  df-v 2754
This theorem is referenced by:  updjud  7112  reumodprminv  12288  grpinveu  12997
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