Theorem caovcand 6039

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 6038	. 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 1240	1 |- ( ph -> ( ( A F B ) = ( A F C ) <-> B = C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148  (class class class)co 5877

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5880

This theorem is referenced by:  caovcanrd  6040  ecopovtrn  6634  ecopovtrng  6637