Theorem subadd

Description: Relationship between subtraction and addition. (Contributed by NM, 20-Jan-1997) (Revised by Mario Carneiro, 21-Dec-2013)

	Ref	Expression
Assertion	subadd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐵 + 𝐶 ) = 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		subval	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 − 𝐵 ) = ( ℩ 𝑥 ∈ ℂ ( 𝐵 + 𝑥 ) = 𝐴 ) )
2	1	eqeq1d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( ℩ 𝑥 ∈ ℂ ( 𝐵 + 𝑥 ) = 𝐴 ) = 𝐶 ) )
3	2	3adant3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( ℩ 𝑥 ∈ ℂ ( 𝐵 + 𝑥 ) = 𝐴 ) = 𝐶 ) )
4		negeu	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ∃! 𝑥 ∈ ℂ ( 𝐵 + 𝑥 ) = 𝐴 )
5		oveq2	⊢ ( 𝑥 = 𝐶 → ( 𝐵 + 𝑥 ) = ( 𝐵 + 𝐶 ) )
6	5	eqeq1d	⊢ ( 𝑥 = 𝐶 → ( ( 𝐵 + 𝑥 ) = 𝐴 ↔ ( 𝐵 + 𝐶 ) = 𝐴 ) )
7	6	riota2	⊢ ( ( 𝐶 ∈ ℂ ∧ ∃! 𝑥 ∈ ℂ ( 𝐵 + 𝑥 ) = 𝐴 ) → ( ( 𝐵 + 𝐶 ) = 𝐴 ↔ ( ℩ 𝑥 ∈ ℂ ( 𝐵 + 𝑥 ) = 𝐴 ) = 𝐶 ) )
8	4 7	sylan2	⊢ ( ( 𝐶 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ) ) → ( ( 𝐵 + 𝐶 ) = 𝐴 ↔ ( ℩ 𝑥 ∈ ℂ ( 𝐵 + 𝑥 ) = 𝐴 ) = 𝐶 ) )
9	8	3impb	⊢ ( ( 𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( ( 𝐵 + 𝐶 ) = 𝐴 ↔ ( ℩ 𝑥 ∈ ℂ ( 𝐵 + 𝑥 ) = 𝐴 ) = 𝐶 ) )
10	9	3com13	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐵 + 𝐶 ) = 𝐴 ↔ ( ℩ 𝑥 ∈ ℂ ( 𝐵 + 𝑥 ) = 𝐴 ) = 𝐶 ) )
11	3 10	bitr4d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐵 + 𝐶 ) = 𝐴 ) )