Theorem caovcanrd 5942

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 5935	. . 3 |- ( ph -> ( A F B ) = ( B F A ) )
5		caovcand.4	. . . 4 |- ( ph -> C e. S )
6	1, 2, 5	caovcomd 5935	. . 3 |- ( ph -> ( A F C ) = ( C F A ) )
7	4, 6	eqeq12d 2155	. 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 5941	. 2 |- ( ph -> ( ( A F B ) = ( A F C ) <-> B = C ) )
11	7, 10	bitr3d 189	1 |- ( ph -> ( ( B F A ) = ( C F A ) <-> B = C ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 963   = wceq 1332   e. wcel 1481  (class class class)co 5782

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 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-iota 5096  df-fv 5139  df-ov 5785

This theorem is referenced by: (None)