Theorem aovnuoveq 32479

Description: The alternative value of the operation on an ordered pair equals the operation's value at this ordered pair. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

Ref	Expression
aovnuoveq	|- ( (( A F B)) =/= _V -> (( A F B)) = ( A F B ) )

Proof of Theorem aovnuoveq

Step	Hyp	Ref	Expression
1		df-aov 32406	. . 3 |- (( A F B)) = ( F''' <. A , B >. )
2	1	neeq1i 2742	. 2 |- ( (( A F B)) =/= _V <-> ( F''' <. A , B >. ) =/= _V )
3		afvnufveq 32435	. . 3 |- ( ( F''' <. A , B >. ) =/= _V -> ( F''' <. A , B >. ) = ( F ` <. A , B >. ) )
4		df-ov 6299	. . 3 |- ( A F B ) = ( F ` <. A , B >. )
5	3, 1, 4	3eqtr4g 2523	. 2 |- ( ( F''' <. A , B >. ) =/= _V -> (( A F B)) = ( A F B ) )
6	2, 5	sylbi 195	1 |- ( (( A F B)) =/= _V -> (( A F B)) = ( A F B ) )

Syntax hints:   -> wi 4   = wceq 1395   =/= wne 2652   _Vcvv 3109   <.cop 4038   ` cfv 5594  (class class class)co 6296  '''cafv 32402   ((caov 32403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-un 3476  df-if 3945  df-fv 5602  df-ov 6299  df-afv 32405  df-aov 32406

This theorem is referenced by: (None)