Theorem xpcan2m 5038

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 5034	. . 3 |- ( E. x x e. C -> ( ( A X. C ) C_ ( B X. C ) <-> A C_ B ) )
2		ssxp1 5034	. . 3 |- ( E. x x e. C -> ( ( B X. C ) C_ ( A X. C ) <-> B C_ A ) )
3	1, 2	anbi12d 465	. 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 3152	. 2 |- ( ( A X. C ) = ( B X. C ) <-> ( ( A X. C ) C_ ( B X. C ) /\ ( B X. C ) C_ ( A X. C ) ) )
5		eqss 3152	. 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 1342   E.wex 1479   e. wcel 2135   C_ wss 3111   X. cxp 4596

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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-br 3977  df-opab 4038  df-xp 4604  df-dm 4608

This theorem is referenced by: (None)