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Metamath Proof Explorer

Theorem addcan2ad

Description: Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	muld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	addcomd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	addcand.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	addcan2ad.4	⊢ ( 𝜑 → ( 𝐴 + 𝐶 ) = ( 𝐵 + 𝐶 ) )
Assertion	addcan2ad	⊢ ( 𝜑 → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		muld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		addcomd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		addcand.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		addcan2ad.4	⊢ ( 𝜑 → ( 𝐴 + 𝐶 ) = ( 𝐵 + 𝐶 ) )
5	1 2 3	addcan2d	⊢ ( 𝜑 → ( ( 𝐴 + 𝐶 ) = ( 𝐵 + 𝐶 ) ↔ 𝐴 = 𝐵 ) )
6	4 5	mpbid	⊢ ( 𝜑 → 𝐴 = 𝐵 )