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Theorem ax11inda 2276
 Description: Induction step for constructing a substitution instance of ax-11o 2217 without using ax-11o 2217. Quantification case. (When and are distinct, ax11inda2 2275 may be used instead to avoid the dummy variable in the proof.) (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ax11inda.1
Assertion
Ref Expression
ax11inda
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   (,,)

Proof of Theorem ax11inda
StepHypRef Expression
1 a9ev 1668 . . 3
2 ax11inda.1 . . . . . . 7
32ax11inda2 2275 . . . . . 6
4 dveeq2-o 2260 . . . . . . . . 9
54imp 419 . . . . . . . 8
6 hba1-o 2225 . . . . . . . . . 10
7 equequ2 1698 . . . . . . . . . . 11
87sps-o 2235 . . . . . . . . . 10
96, 8albidh 1600 . . . . . . . . 9
109notbid 286 . . . . . . . 8
115, 10syl 16 . . . . . . 7
127adantl 453 . . . . . . . 8
138imbi1d 309 . . . . . . . . . . 11
146, 13albidh 1600 . . . . . . . . . 10
155, 14syl 16 . . . . . . . . 9
1615imbi2d 308 . . . . . . . 8
1712, 16imbi12d 312 . . . . . . 7
1811, 17imbi12d 312 . . . . . 6
193, 18mpbii 203 . . . . 5
2019ex 424 . . . 4
2120exlimdv 1646 . . 3
221, 21mpi 17 . 2
2322pm2.43i 45 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wa 359  wal 1549  wex 1550 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-4 2211  ax-5o 2212  ax-6o 2213  ax-10o 2215  ax-12o 2218  ax-16 2220 This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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