Theorem mulcanad

Description: Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcand . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	mulcanad.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	mulcanad.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	mulcanad.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	mulcanad.4	⊢ ( 𝜑 → 𝐶 ≠ 0 )
	mulcanad.5	⊢ ( 𝜑 → ( 𝐶 · 𝐴 ) = ( 𝐶 · 𝐵 ) )
Assertion	mulcanad	⊢ ( 𝜑 → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		mulcanad.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		mulcanad.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		mulcanad.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		mulcanad.4	⊢ ( 𝜑 → 𝐶 ≠ 0 )
5		mulcanad.5	⊢ ( 𝜑 → ( 𝐶 · 𝐴 ) = ( 𝐶 · 𝐵 ) )
6	1 2 3 4	mulcand	⊢ ( 𝜑 → ( ( 𝐶 · 𝐴 ) = ( 𝐶 · 𝐵 ) ↔ 𝐴 = 𝐵 ) )
7	5 6	mpbid	⊢ ( 𝜑 → 𝐴 = 𝐵 )