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Theorem axac3 8515
Description: This theorem asserts that the constant is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 8514 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 8514 . . 3
21ax-gen 1570 . 2
3 dfackm 8217 . 2
42, 3mpbir 202 1
Colors of variables: wff set class
Syntax hints:  -.wn 3  ->wi 4  \/wo 359  /\wa 360  A.wal 1564  E.wex 1565   wac 8167
This theorem is referenced by:  ackm  8516  axac  8518  axaci  8519  cardeqv  8520  fin71ac  8582  lbsex  16860  ptcls  18663  ptcmp  19104  axac10  28530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1570  ax-4 1581  ax-5 1644  ax-6 1685  ax-7 1705  ax-8 1734  ax-9 1736  ax-10 1751  ax-11 1756  ax-12 1768  ax-13 1955  ax-ext 2470  ax-rep 4429  ax-sep 4439  ax-nul 4447  ax-pow 4493  ax-pr 4554  ax-un 6382  ax-ac2 8514
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1338  df-ex 1566  df-nf 1569  df-sb 1677  df-eu 2317  df-mo 2318  df-clab 2476  df-cleq 2482  df-clel 2485  df-nfc 2614  df-ne 2654  df-ral 2764  df-rex 2765  df-reu 2766  df-rab 2768  df-v 3017  df-sbc 3225  df-csb 3326  df-dif 3368  df-un 3370  df-in 3372  df-ss 3379  df-nul 3674  df-if 3826  df-pw 3895  df-sn 3915  df-pr 3916  df-op 3918  df-uni 4118  df-iun 4199  df-br 4319  df-opab 4377  df-mpt 4378  df-id 4657  df-xp 4868  df-rel 4869  df-cnv 4870  df-co 4871  df-dm 4872  df-rn 4873  df-res 4874  df-ima 4875  df-iota 5401  df-fun 5440  df-fn 5441  df-f 5442  df-f1 5443  df-fo 5444  df-f1o 5445  df-fv 5446  df-ac 8168
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