Theorem caovcang 6110

Description: Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

Hypothesis

Ref	Expression
caovcang.1	|- ( ( ph /\ ( x e. T /\ y e. S /\ z e. S ) ) -> ( ( x F y ) = ( x F z ) <-> y = z ) )

Assertion

Ref	Expression
caovcang	|- ( ( ph /\ ( A e. T /\ B e. S /\ C e. S ) ) -> ( ( 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 caovcang

Step	Hyp	Ref	Expression
1		caovcang.1	. . 3 |- ( ( ph /\ ( x e. T /\ y e. S /\ z e. S ) ) -> ( ( x F y ) = ( x F z ) <-> y = z ) )
2	1	ralrimivvva 2589	. 2 |- ( ph -> A. x e. T A. y e. S A. z e. S ( ( x F y ) = ( x F z ) <-> y = z ) )
3		oveq1 5953	. . . . 5 |- ( x = A -> ( x F y ) = ( A F y ) )
4		oveq1 5953	. . . . 5 |- ( x = A -> ( x F z ) = ( A F z ) )
5	3, 4	eqeq12d 2220	. . . 4 |- ( x = A -> ( ( x F y ) = ( x F z ) <-> ( A F y ) = ( A F z ) ) )
6	5	bibi1d 233	. . 3 |- ( x = A -> ( ( ( x F y ) = ( x F z ) <-> y = z ) <-> ( ( A F y ) = ( A F z ) <-> y = z ) ) )
7		oveq2 5954	. . . . 5 |- ( y = B -> ( A F y ) = ( A F B ) )
8	7	eqeq1d 2214	. . . 4 |- ( y = B -> ( ( A F y ) = ( A F z ) <-> ( A F B ) = ( A F z ) ) )
9		eqeq1 2212	. . . 4 |- ( y = B -> ( y = z <-> B = z ) )
10	8, 9	bibi12d 235	. . 3 |- ( y = B -> ( ( ( A F y ) = ( A F z ) <-> y = z ) <-> ( ( A F B ) = ( A F z ) <-> B = z ) ) )
11		oveq2 5954	. . . . 5 |- ( z = C -> ( A F z ) = ( A F C ) )
12	11	eqeq2d 2217	. . . 4 |- ( z = C -> ( ( A F B ) = ( A F z ) <-> ( A F B ) = ( A F C ) ) )
13		eqeq2 2215	. . . 4 |- ( z = C -> ( B = z <-> B = C ) )
14	12, 13	bibi12d 235	. . 3 |- ( z = C -> ( ( ( A F B ) = ( A F z ) <-> B = z ) <-> ( ( A F B ) = ( A F C ) <-> B = C ) ) )
15	6, 10, 14	rspc3v 2893	. 2 |- ( ( A e. T /\ B e. S /\ C e. S ) -> ( A. x e. T A. y e. S A. z e. S ( ( x F y ) = ( x F z ) <-> y = z ) -> ( ( A F B ) = ( A F C ) <-> B = C ) ) )
16	2, 15	mpan9 281	1 |- ( ( ph /\ ( A e. T /\ B e. S /\ C e. S ) ) -> ( ( A F B ) = ( A F C ) <-> B = C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   A.wral 2484  (class class class)co 5946

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-iota 5233  df-fv 5280  df-ov 5949

This theorem is referenced by:  caovcand  6111