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Metamath Proof Explorer

Theorem subneintrd

Description: Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcand . (Contributed by David Moews, 28-Feb-2017)

Ref Expression
Hypotheses negidd.1 ⊢ φ → A ∈ ℂ
pncand.2 ⊢ φ → B ∈ ℂ
subaddd.3 ⊢ φ → C ∈ ℂ
subneintrd.4 ⊢ φ → B ≠ C
Assertion subneintrd ⊢ φ → A − B ≠ A − C

Proof

Step Hyp Ref Expression
1 negidd.1 ⊢ φ → A ∈ ℂ
2 pncand.2 ⊢ φ → B ∈ ℂ
3 subaddd.3 ⊢ φ → C ∈ ℂ
4 subneintrd.4 ⊢ φ → B ≠ C
5 1 2 3 subcanad ⊢ φ → A − B = A − C ↔ B = C
6 5 necon3bid ⊢ φ → A − B ≠ A − C ↔ B ≠ C
7 4 6 mpbird ⊢ φ → A − B ≠ A − C