Theorem caovcanrd 6217

Description: Commute the arguments of an operation cancellation law. (Contributed by Mario Carneiro, 30-Dec-2014.)

Hypotheses

Ref	Expression
caovcang.1	|- ( ( ph /\ ( x e. T /\ y e. S /\ z e. S ) ) -> ( ( x F y ) = ( x F z ) <-> y = z ) )
caovcand.2	|- ( ph -> A e. T )
caovcand.3	|- ( ph -> B e. S )
caovcand.4	|- ( ph -> C e. S )
caovcanrd.5	|- ( ph -> A e. S )
caovcanrd.6	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )

Assertion

Ref	Expression
caovcanrd	|- ( ph -> ( ( B F A ) = ( C F A ) <-> B = C ) )

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

Proof of Theorem caovcanrd

Step	Hyp	Ref	Expression
1		caovcanrd.6	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
2		caovcanrd.5	. . . 4 |- ( ph -> A e. S )
3		caovcand.3	. . . 4 |- ( ph -> B e. S )
4	1, 2, 3	caovcomd 6210	. . 3 |- ( ph -> ( A F B ) = ( B F A ) )
5		caovcand.4	. . . 4 |- ( ph -> C e. S )
6	1, 2, 5	caovcomd 6210	. . 3 |- ( ph -> ( A F C ) = ( C F A ) )
7	4, 6	eqeq12d 2247	. 2 |- ( ph -> ( ( A F B ) = ( A F C ) <-> ( B F A ) = ( C F A ) ) )
8		caovcang.1	. . 3 |- ( ( ph /\ ( x e. T /\ y e. S /\ z e. S ) ) -> ( ( x F y ) = ( x F z ) <-> y = z ) )
9		caovcand.2	. . 3 |- ( ph -> A e. T )
10	8, 9, 3, 5	caovcand 6216	. 2 |- ( ph -> ( ( A F B ) = ( A F C ) <-> B = C ) )
11	7, 10	bitr3d 190	1 |- ( ph -> ( ( B F A ) = ( C F A ) <-> B = C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203  (class class class)co 6049

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-iota 5311  df-fv 5359  df-ov 6052

This theorem is referenced by: (None)