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Theorem fununiq 29127
 Description: The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013.)
Hypotheses
Ref Expression
fununiq.1
fununiq.2
fununiq.3
Assertion
Ref Expression
fununiq

Proof of Theorem fununiq
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dffun2 5604 . 2
2 fununiq.1 . . . 4
3 fununiq.2 . . . 4
4 fununiq.3 . . . 4
5 breq12 4458 . . . . . . . 8
653adant3 1016 . . . . . . 7
7 breq12 4458 . . . . . . . 8
873adant2 1015 . . . . . . 7
96, 8anbi12d 710 . . . . . 6
10 eqeq12 2486 . . . . . . 7
11103adant1 1014 . . . . . 6
129, 11imbi12d 320 . . . . 5
1312spc3gv 3208 . . . 4
142, 3, 4, 13mp3an 1324 . . 3
1514adantl 466 . 2
161, 15sylbi 195 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   w3a 973  wal 1377   wceq 1379   wcel 1767  cvv 3118   class class class wbr 4453   wrel 5010   wfun 5588 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-id 4801  df-cnv 5013  df-co 5014  df-fun 5596 This theorem is referenced by:  funbreq  29128
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