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Theorem ac4c 8340
Description: Equivalent of Axiom of Choice (class version) (Contributed by NM, 10-Feb-1997.)
Hypothesis
Ref Expression
ac4c.1  |-  A  e. 
_V
Assertion
Ref Expression
ac4c  |-  E. f A. x  e.  A  ( x  =/=  (/)  ->  (
f `  x )  e.  x )
Distinct variable group:    x, f, A

Proof of Theorem ac4c
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ac4c.1 . 2  |-  A  e. 
_V
2 raleq 2891 . . 3  |-  ( y  =  A  ->  ( A. x  e.  y 
( x  =/=  (/)  ->  (
f `  x )  e.  x )  <->  A. x  e.  A  ( x  =/=  (/)  ->  ( f `  x )  e.  x
) ) )
32exbidv 1636 . 2  |-  ( y  =  A  ->  ( E. f A. x  e.  y  ( x  =/=  (/)  ->  ( f `  x )  e.  x
)  <->  E. f A. x  e.  A  ( x  =/=  (/)  ->  ( f `  x )  e.  x
) ) )
4 ac4 8339 . 2  |-  E. f A. x  e.  y 
( x  =/=  (/)  ->  (
f `  x )  e.  x )
51, 3, 4vtocl 2993 1  |-  E. f A. x  e.  A  ( x  =/=  (/)  ->  (
f `  x )  e.  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2593   A.wral 2692   _Vcvv 2943   (/)c0 3615   ` cfv 5440
This theorem is referenced by:  axdclem2  8384
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-ac2 8327
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-reu 2699  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ac 7981
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