Theorem xpcan2m 4974

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 4970	. . 3 |- ( E. x x e. C -> ( ( A X. C ) C_ ( B X. C ) <-> A C_ B ) )
2		ssxp1 4970	. . 3 |- ( E. x x e. C -> ( ( B X. C ) C_ ( A X. C ) <-> B C_ A ) )
3	1, 2	anbi12d 464	. 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 3107	. 2 |- ( ( A X. C ) = ( B X. C ) <-> ( ( A X. C ) C_ ( B X. C ) /\ ( B X. C ) C_ ( A X. C ) ) )
5		eqss 3107	. 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 1331   E.wex 1468   e. wcel 1480   C_ wss 3066   X. cxp 4532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-dm 4544

This theorem is referenced by: (None)