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Theorem ax12v 1855
Description: This is a version of ax-12 1854 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. See theorem ax12v2 2083 for the rederivation of ax-c15 2220 from this theorem. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 1837 and ax-13 1999. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
Assertion
Ref Expression
ax12v
Distinct variable group:   ,

Proof of Theorem ax12v
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ax6ev 1749 . 2
2 equequ2 1799 . . . . 5
32biimprd 223 . . . 4
4 ax-5 1704 . . . . . 6
5 ax-12 1854 . . . . . 6
64, 5syl5 32 . . . . 5
73imim1d 75 . . . . . 6
87alimdv 1709 . . . . 5
96, 8syl9r 72 . . . 4
103, 9syld 44 . . 3
1110exlimiv 1722 . 2
121, 11ax-mp 5 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  A.wal 1393  E.wex 1612
This theorem is referenced by:  19.8a  1857  ax12a2  2084  sb56  2172  exsb  2212  mo2v  2289  2eu6  2383  wl-lem-exsb  30015  wl-lem-moexsb  30017  rexsb  32173
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-12 1854
This theorem depends on definitions:  df-bi 185  df-ex 1613
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