Theorem caovcang 5604

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 Fy) = (x Fz) <-> y = z))

Assertion

Ref	Expression
caovcang	|- ((ph /\ (A e. T /\ B e. S /\ C e. S)) -> ((A FB) = (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 Fy) = (x Fz) <-> y = z))
2	1	ralrimivvva 2396	. 2 |- (ph -> A.x e. T A.y e. S A.z e. S ((x Fy) = (x Fz) <-> y = z))
3		oveq1 5462	. . . . 5 |- (x = A -> (x Fy) = (A Fy))
4		oveq1 5462	. . . . 5 |- (x = A -> (x Fz) = (A Fz))
5	3, 4	eqeq12d 2051	. . . 4 |- (x = A -> ((x Fy) = (x Fz) <-> (A Fy) = (A Fz)))
6	5	bibi1d 222	. . 3 |- (x = A -> (((x Fy) = (x Fz) <-> y = z) <-> ((A Fy) = (A Fz) <-> y = z)))
7		oveq2 5463	. . . . 5 |- (y = B -> (A Fy) = (A FB))
8	7	eqeq1d 2045	. . . 4 |- (y = B -> ((A Fy) = (A Fz) <-> (A FB) = (A Fz)))
9		eqeq1 2043	. . . 4 |- (y = B -> (y = z <-> B = z))
10	8, 9	bibi12d 224	. . 3 |- (y = B -> (((A Fy) = (A Fz) <-> y = z) <-> ((A FB) = (A Fz) <-> B = z)))
11		oveq2 5463	. . . . 5 |- (z = C -> (A Fz) = (A F C))
12	11	eqeq2d 2048	. . . 4 |- (z = C -> ((A FB) = (A Fz) <-> (A FB) = (A F C)))
13		eqeq2 2046	. . . 4 |- (z = C -> (B = z <-> B = C))
14	12, 13	bibi12d 224	. . 3 |- (z = C -> (((A FB) = (A Fz) <-> B = z) <-> ((A FB) = (A F C) <-> B = C)))
15	6, 10, 14	rspc3v 2659	. 2 |- ((A e. T /\ B e. S /\ C e. S) -> (A.x e. T A.y e. S A.z e. S ((x Fy) = (x Fz) <-> y = z) -> ((A FB) = (A F C) <-> B = C)))
16	2, 15	mpan9 265	1 |- ((ph /\ (A e. T /\ B e. S /\ C e. S)) -> ((A FB) = (A F C) <-> B = C))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 884   = wceq 1242   e. wcel 1390  A.wral 2300  (class class class)co 5455

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458

This theorem is referenced by:  caovcand  5605