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Theorem sbal1 2204
 Description: A theorem used in elimination of disjoint variable restriction on and by replacing it with a distinctor . (Contributed by NM, 15-May-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2018.)
Assertion
Ref Expression
sbal1
Distinct variable group:   ,

Proof of Theorem sbal1
StepHypRef Expression
1 sb4b 2098 . . . . 5
2 nfnae 2058 . . . . . 6
3 nfeqf2 2041 . . . . . . 7
4 19.21t 1904 . . . . . . . 8
54bicomd 201 . . . . . . 7
63, 5syl 16 . . . . . 6
72, 6albid 1885 . . . . 5
81, 7sylan9bbr 700 . . . 4
9 nfnae 2058 . . . . . . 7
10 sb4b 2098 . . . . . . 7
119, 10albid 1885 . . . . . 6
12 alcom 1845 . . . . . 6
1311, 12syl6bb 261 . . . . 5
1413adantl 466 . . . 4
158, 14bitr4d 256 . . 3
1615ex 434 . 2
17 sbequ12 1992 . . . 4
1817sps 1865 . . 3
19 sbequ12 1992 . . . . 5
2019sps 1865 . . . 4
2120dral2 2066 . . 3
2218, 21bitr3d 255 . 2
2316, 22pm2.61d2 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:  sbal  2206 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
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