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Theorem axac3 8395
 Description: This theorem asserts that the constant is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 8394 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 8394 . . 3
21ax-gen 1556 . 2
3 dfackm 8097 . 2
42, 3mpbir 202 1
 Colors of variables: wff set class Syntax hints:  -.wn 3  ->wi 4  \/wo 359  /\wa 360  A.wal 1550  E.wex 1551   wac 8047 This theorem is referenced by:  ackm  8396  axac  8398  axaci  8399  cardeqv  8400  fin71ac  8462  lbsex  16288  ptcls  17699  ptcmp  18140  axac10  27283 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4354  ax-sep 4364  ax-nul 4372  ax-pow 4416  ax-pr 4442  ax-un 4742  ax-ac2 8394 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-reu 2719  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3766  df-pw 3828  df-sn 3847  df-pr 3848  df-op 3850  df-uni 4044  df-iun 4124  df-br 4244  df-opab 4302  df-mpt 4303  df-id 4539  df-xp 4925  df-rel 4926  df-cnv 4927  df-co 4928  df-dm 4929  df-rn 4930  df-res 4931  df-ima 4932  df-iota 5464  df-fun 5503  df-fn 5504  df-f 5505  df-f1 5506  df-fo 5507  df-f1o 5508  df-fv 5509  df-ac 8048
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