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Theorem ax12inda 32438
 Description: Induction step for constructing a substitution instance of ax-c15 32380 without using ax-c15 32380. Quantification case. (When and are distinct, ax12inda2 32437 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:   ,   ,   ,   ,
Allowed substitution hints:   (,,)

Proof of Theorem ax12inda
StepHypRef Expression
1 ax6ev 1796 . . 3
2 ax12inda.1 . . . . . . 7
32ax12inda2 32437 . . . . . 6
4 dveeq2-o 32423 . . . . . . . . 9
54imp 430 . . . . . . . 8
6 hba1-o 32388 . . . . . . . . . 10
7 equequ2 1849 . . . . . . . . . . 11
87sps-o 32398 . . . . . . . . . 10
96, 8albidh 1720 . . . . . . . . 9
109notbid 295 . . . . . . . 8
115, 10syl 17 . . . . . . 7
127adantl 467 . . . . . . . 8
138imbi1d 318 . . . . . . . . . . 11
146, 13albidh 1720 . . . . . . . . . 10
155, 14syl 17 . . . . . . . . 9
1615imbi2d 317 . . . . . . . 8
1712, 16imbi12d 321 . . . . . . 7
1811, 17imbi12d 321 . . . . . 6
193, 18mpbii 214 . . . . 5
2019ex 435 . . . 4
2120exlimdv 1768 . . 3
221, 21mpi 21 . 2
2322pm2.43i 49 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 187   wa 370  wal 1435  wex 1659 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-c5 32374  ax-c4 32375  ax-c7 32376  ax-c11 32378  ax-c9 32381  ax-c16 32383 This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664 This theorem is referenced by: (None)
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