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Theorem sbidm 1731
 Description: An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
Assertion
Ref Expression
sbidm

Proof of Theorem sbidm
StepHypRef Expression
1 df-sb 1646 . . . . 5
21simplbi 259 . . . 4
32sbimi 1647 . . 3
4 sbequ8 1727 . . 3
53, 4sylibr 137 . 2
6 ax-1 5 . . 3
7 sb1 1649 . . . 4
8 pm4.24 375 . . . . . . . 8
9 ax-ie1 1382 . . . . . . . . 9
10919.41h 1575 . . . . . . . 8
118, 10bitr4i 176 . . . . . . 7
12 ax-1 5 . . . . . . . . . 10
1312anim2i 324 . . . . . . . . 9
1413anim1i 323 . . . . . . . 8
1514eximi 1491 . . . . . . 7
1611, 15sylbi 114 . . . . . 6
17 anass 381 . . . . . . 7
1817exbii 1496 . . . . . 6
1916, 18sylib 127 . . . . 5
201anbi2i 430 . . . . . 6
2120exbii 1496 . . . . 5
2219, 21sylibr 137 . . . 4
237, 22syl 14 . . 3
24 df-sb 1646 . . 3
256, 23, 24sylanbrc 394 . 2
265, 25impbii 117 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98  wex 1381  wsb 1645 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-sb 1646 This theorem is referenced by: (None)
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