Theorem caovcand 6004

Description: Convert an operation cancellation law to class notation. (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 )

Assertion

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

Step	Hyp	Ref	Expression
1		id 19	. 2 |- ( ph -> ph )
2		caovcand.2	. 2 |- ( ph -> A e. T )
3		caovcand.3	. 2 |- ( ph -> B e. S )
4		caovcand.4	. 2 |- ( ph -> C e. S )
5		caovcang.1	. . 3 |- ( ( ph /\ ( x e. T /\ y e. S /\ z e. S ) ) -> ( ( x F y ) = ( x F z ) <-> y = z ) )
6	5	caovcang 6003	. 2 |- ( ( ph /\ ( A e. T /\ B e. S /\ C e. S ) ) -> ( ( A F B ) = ( A F C ) <-> B = C ) )
7	1, 2, 3, 4, 6	syl13anc 1230	1 |- ( ph -> ( ( A F B ) = ( A F C ) <-> B = C ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136  (class class class)co 5842

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845

This theorem is referenced by:  caovcanrd  6005  ecopovtrn  6598  ecopovtrng  6601