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Theorem ax11inda 2200
 Description: Induction step for constructing a substitution instance of ax-11o 2141 without using ax-11o 2141. Quantification case. (When and are distinct, ax11inda2 2199 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 1656 . . 3
2 ax11inda.1 . . . . . . 7
32ax11inda2 2199 . . . . . 6
4 dveeq2-o 2184 . . . . . . . . 9
54imp 418 . . . . . . . 8
6 hba1-o 2149 . . . . . . . . . 10
7 equequ2 1686 . . . . . . . . . . 11
87sps-o 2159 . . . . . . . . . 10
96, 8albidh 1590 . . . . . . . . 9
109notbid 285 . . . . . . . 8
115, 10syl 15 . . . . . . 7
127adantl 452 . . . . . . . 8
138imbi1d 308 . . . . . . . . . . 11
146, 13albidh 1590 . . . . . . . . . 10
155, 14syl 15 . . . . . . . . 9
1615imbi2d 307 . . . . . . . 8
1712, 16imbi12d 311 . . . . . . 7
1811, 17imbi12d 311 . . . . . 6
193, 18mpbii 202 . . . . 5
2019ex 423 . . . 4
2120exlimdv 1636 . . 3
221, 21mpi 16 . 2
2322pm2.43i 43 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 176   wa 358  wal 1540  wex 1541 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-10o 2139  ax-12o 2142  ax-16 2144 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545 This theorem is referenced by: (None)
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