Theorem caovcanrd 6837

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 6830	. . 3 |- ( ph -> ( A F B ) = ( B F A ) )
5		caovcand.4	. . . 4 |- ( ph -> C e. S )
6	1, 2, 5	caovcomd 6830	. . 3 |- ( ph -> ( A F C ) = ( C F A ) )
7	4, 6	eqeq12d 2637	. 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 6836	. 2 |- ( ph -> ( ( A F B ) = ( A F C ) <-> B = C ) )
11	7, 10	bitr3d 270	1 |- ( ph -> ( ( B F A ) = ( C F A ) <-> B = C ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by: (None)