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Theorem ax11inda 1819
Description: Induction step for constructing a substitution instance of ax-11o 1491 without using ax-11o 1491. Quantification case. (When and are distinct, ax11inda2 1818 may be used instead to avoid the dummy variable in the proof.) (Contributed by NM, 24-Jan-2007.)
Hypothesis
Ref Expression
ax11inda.1
Assertion
Ref Expression
ax11inda
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   (,,)

Proof of Theorem ax11inda
StepHypRef Expression
1 a9e 1389 . . 3
2 ax11inda.1 . . . . . . 7
32ax11inda2 1818 . . . . . 6
4 dveeq2 1483 . . . . . . . . 9
54imp 113 . . . . . . . 8
6 hba1 1299 . . . . . . . . . 10
7 equequ2 1401 . . . . . . . . . . 11
87a4s 1296 . . . . . . . . . 10
96, 8albid 1239 . . . . . . . . 9
109notbid 570 . . . . . . . 8
115, 10syl 13 . . . . . . 7
127adantl 260 . . . . . . . 8
138imbi1d 218 . . . . . . . . . . 11
146, 13albid 1239 . . . . . . . . . 10
155, 14syl 13 . . . . . . . . 9
1615imbi2d 217 . . . . . . . 8
1712, 16imbi12d 221 . . . . . . 7
1811, 17imbi12d 221 . . . . . 6
193, 18mpbii 134 . . . . 5
2019ex 106 . . . 4
2120exlimdv 1487 . . 3
221, 21mpi 14 . 2
2322pm2.43i 41 1
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 95   wb 96  wal 1214  wex 1253
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 97  ax-ia2 98  ax-ia3 99  ax-in1 526  ax-in2 527  ax-io 606  ax-3 714  ax-5 1215  ax-6 1216  ax-7 1217  ax-gen 1218  ax-ie1 1254  ax-ie2 1255  ax-8 1266  ax-10 1267  ax-11 1268  ax-i12 1270  ax-4 1271  ax-17 1280  ax-i9 1282  ax-ial 1293  ax-i5r 1294  ax-16 1481
This theorem depends on definitions:  df-bi 108
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