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Theorem axac3 8458
Description: This theorem asserts that the constant is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 8457 as an axiom. (Contributed by Mario Carneiro, 6-May-2015.) (Revised by NM, 20-Dec-2016.) (Proof modification is discouraged.)
Assertion
Ref Expression
axac3

Proof of Theorem axac3
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-ac2 8457 . . 3
21ax-gen 1562 . 2
3 dfackm 8160 . 2
42, 3mpbir 202 1
Colors of variables: wff set class
Syntax hints:  -.wn 3  ->wi 4  \/wo 359  /\wa 360  A.wal 1556  E.wex 1557   wac 8110
This theorem is referenced by:  ackm  8459  axac  8461  axaci  8462  cardeqv  8463  fin71ac  8525  lbsex  16740  ptcls  18152  ptcmp  18593  axac10  28033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1562  ax-4 1573  ax-5 1636  ax-6 1677  ax-7 1697  ax-8 1726  ax-9 1728  ax-10 1743  ax-11 1748  ax-12 1760  ax-13 1947  ax-ext 2462  ax-rep 4413  ax-sep 4423  ax-nul 4431  ax-pow 4477  ax-pr 4538  ax-un 6338  ax-ac2 8457
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1337  df-ex 1558  df-nf 1561  df-sb 1669  df-eu 2309  df-mo 2310  df-clab 2468  df-cleq 2474  df-clel 2477  df-nfc 2606  df-ne 2646  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3008  df-sbc 3213  df-csb 3314  df-dif 3356  df-un 3358  df-in 3360  df-ss 3367  df-nul 3661  df-if 3813  df-pw 3880  df-sn 3900  df-pr 3901  df-op 3903  df-uni 4102  df-iun 4183  df-br 4303  df-opab 4361  df-mpt 4362  df-id 4639  df-xp 4850  df-rel 4851  df-cnv 4852  df-co 4853  df-dm 4854  df-rn 4855  df-res 4856  df-ima 4857  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ac 8111
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