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Metamath Proof Explorer

Theorem mulcan2ad

Description: Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcan2d . (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 )
	mulcan2ad.5	⊢ ( 𝜑 → ( 𝐴 · 𝐶 ) = ( 𝐵 · 𝐶 ) )
Assertion	mulcan2ad	⊢ ( 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		mulcan2ad.5	⊢ ( 𝜑 → ( 𝐴 · 𝐶 ) = ( 𝐵 · 𝐶 ) )
6	1 2 3 4	mulcan2d	⊢ ( 𝜑 → ( ( 𝐴 · 𝐶 ) = ( 𝐵 · 𝐶 ) ↔ 𝐴 = 𝐵 ) )
7	5 6	mpbid	⊢ ( 𝜑 → 𝐴 = 𝐵 )