Theorem caovcanrd 5934

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 5927	. . 3 |- ( ph -> ( A F B ) = ( B F A ) )
5		caovcand.4	. . . 4 |- ( ph -> C e. S )
6	1, 2, 5	caovcomd 5927	. . 3 |- ( ph -> ( A F C ) = ( C F A ) )
7	4, 6	eqeq12d 2154	. 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 5933	. 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 962   = wceq 1331   e. wcel 1480  (class class class)co 5774

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777

This theorem is referenced by: (None)