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Metamath Proof Explorer

Theorem pncan

Description: Cancellation law for subtraction. (Contributed by NM, 10-May-2004) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion pncan 
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - B ) = A )

Proof

Step Hyp Ref Expression
1 simpr 
 |-  ( ( A e. CC /\ B e. CC ) -> B e. CC )
2 simpl 
 |-  ( ( A e. CC /\ B e. CC ) -> A e. CC )
3 1 2 addcomd 
 |-  ( ( A e. CC /\ B e. CC ) -> ( B + A ) = ( A + B ) )
4 addcl 
 |-  ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
5 subadd 
 |-  ( ( ( A + B ) e. CC /\ B e. CC /\ A e. CC ) -> ( ( ( A + B ) - B ) = A <-> ( B + A ) = ( A + B ) ) )
6 4 1 2 5 syl3anc 
 |-  ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) - B ) = A <-> ( B + A ) = ( A + B ) ) )
7 3 6 mpbird 
 |-  ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - B ) = A )