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Theorem ax12inda 2278
 Description: Induction step for constructing a substitution instance of ax-c15 2220 without using ax-c15 2220. Quantification case. (When and are distinct, ax12inda2 2277 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
ax12inda.1
Assertion
Ref Expression
ax12inda
Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem ax12inda
StepHypRef Expression
1 ax6ev 1749 . . 3
2 ax12inda.1 . . . . . . 7
32ax12inda2 2277 . . . . . 6
4 dveeq2-o 2263 . . . . . . . . 9
54imp 429 . . . . . . . 8
6 hba1-o 2228 . . . . . . . . . 10
7 equequ2 1799 . . . . . . . . . . 11
87sps-o 2238 . . . . . . . . . 10
96, 8albidh 1675 . . . . . . . . 9
109notbid 294 . . . . . . . 8
115, 10syl 16 . . . . . . 7
127adantl 466 . . . . . . . 8
138imbi1d 317 . . . . . . . . . . 11
146, 13albidh 1675 . . . . . . . . . 10
155, 14syl 16 . . . . . . . . 9
1615imbi2d 316 . . . . . . . 8
1712, 16imbi12d 320 . . . . . . 7
1811, 17imbi12d 320 . . . . . 6
193, 18mpbii 211 . . . . 5
2019ex 434 . . . 4
2120exlimdv 1724 . . 3
221, 21mpi 17 . 2
2322pm2.43i 47 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612 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  ax-c5 2214  ax-c4 2215  ax-c7 2216  ax-c11 2218  ax-c9 2221  ax-c16 2223 This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617
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