MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sb8eu Unicode version

Theorem sb8eu 2318
Description: Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.)
Hypothesis
Ref Expression
sb8eu.1
Assertion
Ref Expression
sb8eu

Proof of Theorem sb8eu
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1707 . . . . 5
21sb8 2167 . . . 4
3 equsb3 2176 . . . . . 6
43sblbis 2145 . . . . 5
54albii 1640 . . . 4
6 sb8eu.1 . . . . . . 7
76nfsb 2184 . . . . . 6
8 nfv 1707 . . . . . 6
97, 8nfbi 1934 . . . . 5
10 nfv 1707 . . . . 5
11 sbequ 2117 . . . . . 6
12 equequ1 1798 . . . . . 6
1311, 12bibi12d 321 . . . . 5
149, 10, 13cbval 2021 . . . 4
152, 5, 143bitri 271 . . 3
1615exbii 1667 . 2
17 df-eu 2286 . 2
18 df-eu 2286 . 2
1916, 17, 183bitr4i 277 1
Colors of variables: wff setvar class
Syntax hints:  <->wb 184  A.wal 1393  E.wex 1612  F/wnf 1616  [wsb 1739  E!weu 2282
This theorem is referenced by:  sb8mo  2320  cbveu  2321  eu1  2327  cbvreu  3082
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
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
  Copyright terms: Public domain W3C validator