Theorem xpcan2m 5106

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 5102	. . 3 |- ( E. x x e. C -> ( ( A X. C ) C_ ( B X. C ) <-> A C_ B ) )
2		ssxp1 5102	. . 3 |- ( E. x x e. C -> ( ( B X. C ) C_ ( A X. C ) <-> B C_ A ) )
3	1, 2	anbi12d 473	. 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 3194	. 2 |- ( ( A X. C ) = ( B X. C ) <-> ( ( A X. C ) C_ ( B X. C ) /\ ( B X. C ) C_ ( A X. C ) ) )
5		eqss 3194	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
6	3, 4, 5	3bitr4g 223	1 |- ( E. x x e. C -> ( ( A X. C ) = ( B X. C ) <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1503   e. wcel 2164   C_ wss 3153   X. cxp 4657

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-dm 4669

This theorem is referenced by: (None)