Metamath Proof Explorer


Theorem mulcan2ad

Description: Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcan2d . (Contributed by David Moews, 28-Feb-2017)

Ref Expression
Hypotheses mulcanad.1
|- ( ph -> A e. CC )
mulcanad.2
|- ( ph -> B e. CC )
mulcanad.3
|- ( ph -> C e. CC )
mulcanad.4
|- ( ph -> C =/= 0 )
mulcan2ad.5
|- ( ph -> ( A x. C ) = ( B x. C ) )
Assertion mulcan2ad
|- ( ph -> A = B )

Proof

Step Hyp Ref Expression
1 mulcanad.1
 |-  ( ph -> A e. CC )
2 mulcanad.2
 |-  ( ph -> B e. CC )
3 mulcanad.3
 |-  ( ph -> C e. CC )
4 mulcanad.4
 |-  ( ph -> C =/= 0 )
5 mulcan2ad.5
 |-  ( ph -> ( A x. C ) = ( B x. C ) )
6 1 2 3 4 mulcan2d
 |-  ( ph -> ( ( A x. C ) = ( B x. C ) <-> A = B ) )
7 5 6 mpbid
 |-  ( ph -> A = B )