Theorem divne1d

Description: If two complex numbers are unequal, their quotient is not one. Contrapositive of diveq1d . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	div1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	divcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	divcld.3	⊢ ( 𝜑 → 𝐵 ≠ 0 )
	divne1d.4	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
Assertion	divne1d	⊢ ( 𝜑 → ( 𝐴 / 𝐵 ) ≠ 1 )

Proof

Step	Hyp	Ref	Expression
1		div1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		divcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		divcld.3	⊢ ( 𝜑 → 𝐵 ≠ 0 )
4		divne1d.4	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
5	1 2 3	diveq1ad	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) = 1 ↔ 𝐴 = 𝐵 ) )
6	5	necon3bid	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) ≠ 1 ↔ 𝐴 ≠ 𝐵 ) )
7	4 6	mpbird	⊢ ( 𝜑 → ( 𝐴 / 𝐵 ) ≠ 1 )