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Theorem aovovn0oveq 27989
Description: If the operation's value at an argument is not the empty set, it equals the value of the alternative operation at this argument. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
aovovn0oveq  |-  ( ( A F B )  =/=  (/)  -> (( A F B))  =  ( A F B ) )

Proof of Theorem aovovn0oveq
StepHypRef Expression
1 df-ov 6076 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
21neeq1i 2608 . 2  |-  ( ( A F B )  =/=  (/)  <->  ( F `  <. A ,  B >. )  =/=  (/) )
3 afvfvn0fveq 27945 . . 3  |-  ( ( F `  <. A ,  B >. )  =/=  (/)  ->  ( F'''
<. A ,  B >. )  =  ( F `  <. A ,  B >. ) )
4 df-aov 27907 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
53, 4, 13eqtr4g 2492 . 2  |-  ( ( F `  <. A ,  B >. )  =/=  (/)  -> (( A F B))  =  ( A F B ) )
62, 5sylbi 188 1  |-  ( ( A F B )  =/=  (/)  -> (( A F B))  =  ( A F B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    =/= wne 2598   (/)c0 3620   <.cop 3809   ` cfv 5446  (class class class)co 6073  '''cafv 27903   ((caov 27904
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-dfat 27905  df-afv 27906  df-aov 27907
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