Theorem caovcang 6050

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 2570	. 2 |- ( ph -> A. x e. T A. y e. S A. z e. S ( ( x F y ) = ( x F z ) <-> y = z ) )
3		oveq1 5895	. . . . 5 |- ( x = A -> ( x F y ) = ( A F y ) )
4		oveq1 5895	. . . . 5 |- ( x = A -> ( x F z ) = ( A F z ) )
5	3, 4	eqeq12d 2202	. . . 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 5896	. . . . 5 |- ( y = B -> ( A F y ) = ( A F B ) )
8	7	eqeq1d 2196	. . . 4 |- ( y = B -> ( ( A F y ) = ( A F z ) <-> ( A F B ) = ( A F z ) ) )
9		eqeq1 2194	. . . 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 5896	. . . . 5 |- ( z = C -> ( A F z ) = ( A F C ) )
12	11	eqeq2d 2199	. . . 4 |- ( z = C -> ( ( A F B ) = ( A F z ) <-> ( A F B ) = ( A F C ) ) )
13		eqeq2 2197	. . . 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 2869	. 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 979   = wceq 1363   e. wcel 2158   A.wral 2465  (class class class)co 5888

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236  df-ov 5891

This theorem is referenced by:  caovcand  6051