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

Theorem sbralie 3097
Description: Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)
Hypothesis
Ref Expression
sbralie.1
Assertion
Ref Expression
sbralie
Distinct variable groups:   ,   ,   ,

Proof of Theorem sbralie
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 cbvralsv 3095 . . . 4
21sbbii 1746 . . 3
3 nfv 1707 . . . 4
4 raleq 3054 . . . 4
53, 4sbie 2149 . . 3
62, 5bitri 249 . 2
7 cbvralsv 3095 . . 3
8 nfv 1707 . . . . . 6
98sbco2 2158 . . . . 5
10 nfv 1707 . . . . . 6
11 sbralie.1 . . . . . 6
1210, 11sbie 2149 . . . . 5
139, 12bitri 249 . . . 4
1413ralbii 2888 . . 3
157, 14bitri 249 . 2
166, 15bitri 249 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  [wsb 1739  A.wral 2807
This theorem is referenced by:  tfinds2  6698
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-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812
  Copyright terms: Public domain W3C validator