Theorem xpcan2m 4761

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 4757	. . 3 |- ( E. x x e. C -> ( ( A X. C ) C_ ( B X. C ) <-> A C_ B ) )
2		ssxp1 4757	. . 3 |- ( E. x x e. C -> ( ( B X. C ) C_ ( A X. C ) <-> B C_ A ) )
3	1, 2	anbi12d 442	. 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 2960	. 2 |- ( ( A X. C ) = ( B X. C ) <-> ( ( A X. C ) C_ ( B X. C ) /\ ( B X. C ) C_ ( A X. C ) ) )
5		eqss 2960	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
6	3, 4, 5	3bitr4g 212	1 |- ( E. x x e. C -> ( ( A X. C ) = ( B X. C ) <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   C_ wss 2917   X. cxp 4343

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-dm 4355

This theorem is referenced by: (None)