Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvreu Unicode version

Theorem cbvreu 3082
 Description: Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
cbvral.1
cbvral.2
cbvral.3
Assertion
Ref Expression
cbvreu
Distinct variable groups:   ,   ,

Proof of Theorem cbvreu
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1707 . . . 4
21sb8eu 2318 . . 3
3 sban 2140 . . . 4
43eubii 2306 . . 3
5 clelsb3 2578 . . . . . 6
65anbi1i 695 . . . . 5
76eubii 2306 . . . 4
8 nfv 1707 . . . . . 6
9 cbvral.1 . . . . . . 7
109nfsb 2184 . . . . . 6
118, 10nfan 1928 . . . . 5
12 nfv 1707 . . . . 5
13 eleq1 2529 . . . . . 6
14 sbequ 2117 . . . . . . 7
15 cbvral.2 . . . . . . . 8
16 cbvral.3 . . . . . . . 8
1715, 16sbie 2149 . . . . . . 7
1814, 17syl6bb 261 . . . . . 6
1913, 18anbi12d 710 . . . . 5
2011, 12, 19cbveu 2321 . . . 4
217, 20bitri 249 . . 3
222, 4, 213bitri 271 . 2
23 df-reu 2814 . 2
24 df-reu 2814 . 2
2522, 23, 243bitr4i 277 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  F/wnf 1616  [wsb 1739  e.wcel 1818  E!weu 2282  E!wreu 2809 This theorem is referenced by:  cbvrmo  3083  cbvreuv  3086 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-cleq 2449  df-clel 2452  df-reu 2814
 Copyright terms: Public domain W3C validator