Theorem caovcanrd 6133

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 6126	. . 3 |- ( ph -> ( A F B ) = ( B F A ) )
5		caovcand.4	. . . 4 |- ( ph -> C e. S )
6	1, 2, 5	caovcomd 6126	. . 3 |- ( ph -> ( A F C ) = ( C F A ) )
7	4, 6	eqeq12d 2222	. 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 6132	. 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 981   = wceq 1373   e. wcel 2178  (class class class)co 5967

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-iota 5251  df-fv 5298  df-ov 5970

This theorem is referenced by: (None)