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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 ) )