Theorem caovcan 5975

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 5821	. . . 4 |- ( x = A -> ( x F y ) = ( A F y ) )
2		oveq1 5821	. . . 4 |- ( x = A -> ( x F C ) = ( A F C ) )
3	1, 2	eqeq12d 2169	. . 3 |- ( x = A -> ( ( x F y ) = ( x F C ) <-> ( A F y ) = ( A F C ) ) )
4	3	imbi1d 230	. 2 |- ( x = A -> ( ( ( x F y ) = ( x F C ) -> y = C ) <-> ( ( A F y ) = ( A F C ) -> y = C ) ) )
5		oveq2 5822	. . . 4 |- ( y = B -> ( A F y ) = ( A F B ) )
6	5	eqeq1d 2163	. . 3 |- ( y = B -> ( ( A F y ) = ( A F C ) <-> ( A F B ) = ( A F C ) ) )
7		eqeq1 2161	. . 3 |- ( y = B -> ( y = C <-> B = C ) )
8	6, 7	imbi12d 233	. 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 5822	. . . . . 6 |- ( z = C -> ( x F z ) = ( x F C ) )
11	10	eqeq2d 2166	. . . . 5 |- ( z = C -> ( ( x F y ) = ( x F z ) <-> ( x F y ) = ( x F C ) ) )
12		eqeq2 2164	. . . . 5 |- ( z = C -> ( y = z <-> y = C ) )
13	11, 12	imbi12d 233	. . . 4 |- ( z = C -> ( ( ( x F y ) = ( x F z ) -> y = z ) <-> ( ( x F y ) = ( x F C ) -> y = C ) ) )
14	13	imbi2d 229	. . 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 2763	. 2 |- ( ( x e. S /\ y e. S ) -> ( ( x F y ) = ( x F C ) -> y = C ) )
17	4, 8, 16	vtocl2ga 2777	1 |- ( ( A e. S /\ B e. S ) -> ( ( A F B ) = ( A F C ) -> B = C ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 2125   _Vcvv 2709  (class class class)co 5814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-rex 2438  df-v 2711  df-un 3102  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-iota 5128  df-fv 5171  df-ov 5817

This theorem is referenced by: (None)