Theorem caovcan 5665

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 5519	. . . 4 |- ( x = A -> ( x F y ) = ( A F y ) )
2		oveq1 5519	. . . 4 |- ( x = A -> ( x F C ) = ( A F C ) )
3	1, 2	eqeq12d 2054	. . 3 |- ( x = A -> ( ( x F y ) = ( x F C ) <-> ( A F y ) = ( A F C ) ) )
4	3	imbi1d 220	. 2 |- ( x = A -> ( ( ( x F y ) = ( x F C ) -> y = C ) <-> ( ( A F y ) = ( A F C ) -> y = C ) ) )
5		oveq2 5520	. . . 4 |- ( y = B -> ( A F y ) = ( A F B ) )
6	5	eqeq1d 2048	. . 3 |- ( y = B -> ( ( A F y ) = ( A F C ) <-> ( A F B ) = ( A F C ) ) )
7		eqeq1 2046	. . 3 |- ( y = B -> ( y = C <-> B = C ) )
8	6, 7	imbi12d 223	. 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 5520	. . . . . 6 |- ( z = C -> ( x F z ) = ( x F C ) )
11	10	eqeq2d 2051	. . . . 5 |- ( z = C -> ( ( x F y ) = ( x F z ) <-> ( x F y ) = ( x F C ) ) )
12		eqeq2 2049	. . . . 5 |- ( z = C -> ( y = z <-> y = C ) )
13	11, 12	imbi12d 223	. . . 4 |- ( z = C -> ( ( ( x F y ) = ( x F z ) -> y = z ) <-> ( ( x F y ) = ( x F C ) -> y = C ) ) )
14	13	imbi2d 219	. . 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 2608	. 2 |- ( ( x e. S /\ y e. S ) -> ( ( x F y ) = ( x F C ) -> y = C ) )
17	4, 8, 16	vtocl2ga 2621	1 |- ( ( A e. S /\ B e. S ) -> ( ( A F B ) = ( A F C ) -> B = C ) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   _Vcvv 2557  (class class class)co 5512

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515

This theorem is referenced by: (None)