Theorem xpcan2m 4884

Description: Cancellation law for cross-product. (Contributed by Jim Kingdon, 14-Dec-2018.)

Assertion

Ref	Expression
xpcan2m	|- ( E. x x e. C -> ( ( A X. C ) = ( B X. C ) <-> A = B ) )

Distinct variable group:   x, C

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem xpcan2m

Step	Hyp	Ref	Expression
1		ssxp1 4880	. . 3 |- ( E. x x e. C -> ( ( A X. C ) C_ ( B X. C ) <-> A C_ B ) )
2		ssxp1 4880	. . 3 |- ( E. x x e. C -> ( ( B X. C ) C_ ( A X. C ) <-> B C_ A ) )
3	1, 2	anbi12d 458	. 2 |- ( E. x x e. C -> ( ( ( A X. C ) C_ ( B X. C ) /\ ( B X. C ) C_ ( A X. C ) ) <-> ( A C_ B /\ B C_ A ) ) )
4		eqss 3041	. 2 |- ( ( A X. C ) = ( B X. C ) <-> ( ( A X. C ) C_ ( B X. C ) /\ ( B X. C ) C_ ( A X. C ) ) )
5		eqss 3041	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
6	3, 4, 5	3bitr4g 222	1 |- ( E. x x e. C -> ( ( A X. C ) = ( B X. C ) <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1290   E.wex 1427   e. wcel 1439   C_ wss 3000   X. cxp 4450

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4458  df-dm 4462

This theorem is referenced by: (None)