MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbal2 Unicode version

Theorem sbal2 2205
Description: Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.) Remove a distinct variable constraint. (Revised by Wolf Lammen, 3-Oct-2018.)
Assertion
Ref Expression
sbal2
Distinct variable group:   ,

Proof of Theorem sbal2
StepHypRef Expression
1 sb4b 2098 . . . . 5
21adantl 466 . . . 4
3 nfnae 2058 . . . . . 6
4 sb4b 2098 . . . . . 6
53, 4albid 1885 . . . . 5
6 alcom 1845 . . . . . 6
7 nfnae 2058 . . . . . . 7
8 nfeqf1 2043 . . . . . . . 8
9 19.21t 1904 . . . . . . . 8
108, 9syl 16 . . . . . . 7
117, 10albid 1885 . . . . . 6
126, 11syl5bb 257 . . . . 5
135, 12sylan9bbr 700 . . . 4
142, 13bitr4d 256 . . 3
1514ex 434 . 2
16 sbid 1996 . . . 4
17 drsb2 2119 . . . 4
1816, 17syl5bbr 259 . . 3
19 sbid 1996 . . . . 5
20 drsb2 2119 . . . . 5
2119, 20syl5bbr 259 . . . 4
2221dral2 2066 . . 3
2318, 22bitr3d 255 . 2
2415, 23pm2.61d2 160 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  F/wnf 1616  [wsb 1739
This theorem is referenced by:  2sb5ndVD  33710  2sb5ndALT  33732
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-an 371  df-ex 1613  df-nf 1617  df-sb 1740
  Copyright terms: Public domain W3C validator