Metamath Proof Explorer


Theorem addneintrd

Description: Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad . Consequence of addcand . (Contributed by David Moews, 28-Feb-2017)

Ref Expression
Hypotheses muld.1 ( 𝜑𝐴 ∈ ℂ )
addcomd.2 ( 𝜑𝐵 ∈ ℂ )
addcand.3 ( 𝜑𝐶 ∈ ℂ )
addneintrd.4 ( 𝜑𝐵𝐶 )
Assertion addneintrd ( 𝜑 → ( 𝐴 + 𝐵 ) ≠ ( 𝐴 + 𝐶 ) )

Proof

Step Hyp Ref Expression
1 muld.1 ( 𝜑𝐴 ∈ ℂ )
2 addcomd.2 ( 𝜑𝐵 ∈ ℂ )
3 addcand.3 ( 𝜑𝐶 ∈ ℂ )
4 addneintrd.4 ( 𝜑𝐵𝐶 )
5 1 2 3 addcand ( 𝜑 → ( ( 𝐴 + 𝐵 ) = ( 𝐴 + 𝐶 ) ↔ 𝐵 = 𝐶 ) )
6 5 necon3bid ( 𝜑 → ( ( 𝐴 + 𝐵 ) ≠ ( 𝐴 + 𝐶 ) ↔ 𝐵𝐶 ) )
7 4 6 mpbird ( 𝜑 → ( 𝐴 + 𝐵 ) ≠ ( 𝐴 + 𝐶 ) )