Theorem caovcan 6041

Description: Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.)

Hypotheses

Ref	Expression
caovcan.1	|- C e. _V
caovcan.2	|- ( ( x e. S /\ y e. S ) -> ( ( x F y ) = ( x F z ) -> y = z ) )

Assertion

Ref	Expression
caovcan	|- ( ( A e. S /\ B e. S ) -> ( ( A F B ) = ( A F C ) -> B = C ) )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, C, y, z   x, F, y, z   x, S, y, z

Proof of Theorem caovcan

Step	Hyp	Ref	Expression
1		oveq1 5884	. . . 4 |- ( x = A -> ( x F y ) = ( A F y ) )
2		oveq1 5884	. . . 4 |- ( x = A -> ( x F C ) = ( A F C ) )
3	1, 2	eqeq12d 2192	. . 3 |- ( x = A -> ( ( x F y ) = ( x F C ) <-> ( A F y ) = ( A F C ) ) )
4	3	imbi1d 231	. 2 |- ( x = A -> ( ( ( x F y ) = ( x F C ) -> y = C ) <-> ( ( A F y ) = ( A F C ) -> y = C ) ) )
5		oveq2 5885	. . . 4 |- ( y = B -> ( A F y ) = ( A F B ) )
6	5	eqeq1d 2186	. . 3 |- ( y = B -> ( ( A F y ) = ( A F C ) <-> ( A F B ) = ( A F C ) ) )
7		eqeq1 2184	. . 3 |- ( y = B -> ( y = C <-> B = C ) )
8	6, 7	imbi12d 234	. 2 |- ( y = B -> ( ( ( A F y ) = ( A F C ) -> y = C ) <-> ( ( A F B ) = ( A F C ) -> B = C ) ) )
9		caovcan.1	. . 3 |- C e. _V
10		oveq2 5885	. . . . . 6 |- ( z = C -> ( x F z ) = ( x F C ) )
11	10	eqeq2d 2189	. . . . 5 |- ( z = C -> ( ( x F y ) = ( x F z ) <-> ( x F y ) = ( x F C ) ) )
12		eqeq2 2187	. . . . 5 |- ( z = C -> ( y = z <-> y = C ) )
13	11, 12	imbi12d 234	. . . 4 |- ( z = C -> ( ( ( x F y ) = ( x F z ) -> y = z ) <-> ( ( x F y ) = ( x F C ) -> y = C ) ) )
14	13	imbi2d 230	. . 3 |- ( z = C -> ( ( ( x e. S /\ y e. S ) -> ( ( x F y ) = ( x F z ) -> y = z ) ) <-> ( ( x e. S /\ y e. S ) -> ( ( x F y ) = ( x F C ) -> y = C ) ) ) )
15		caovcan.2	. . 3 |- ( ( x e. S /\ y e. S ) -> ( ( x F y ) = ( x F z ) -> y = z ) )
16	9, 14, 15	vtocl 2793	. 2 |- ( ( x e. S /\ y e. S ) -> ( ( x F y ) = ( x F C ) -> y = C ) )
17	4, 8, 16	vtocl2ga 2807	1 |- ( ( A e. S /\ B e. S ) -> ( ( A F B ) = ( A F C ) -> B = C ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   _Vcvv 2739  (class class class)co 5877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5880

This theorem is referenced by: (None)