Theorem xneg11

Description: Extended real version of neg11 . (Contributed by Mario Carneiro, 20-Aug-2015)

	Ref	Expression
Assertion	xneg11	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( -𝑒 𝐴 = -𝑒 𝐵 ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		xnegeq	⊢ ( -𝑒 𝐴 = -𝑒 𝐵 → -𝑒 -𝑒 𝐴 = -𝑒 -𝑒 𝐵 )
2		xnegneg	⊢ ( 𝐴 ∈ ℝ* → -𝑒 -𝑒 𝐴 = 𝐴 )
3		xnegneg	⊢ ( 𝐵 ∈ ℝ* → -𝑒 -𝑒 𝐵 = 𝐵 )
4	2 3	eqeqan12d	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( -𝑒 -𝑒 𝐴 = -𝑒 -𝑒 𝐵 ↔ 𝐴 = 𝐵 ) )
5	1 4	syl5ib	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( -𝑒 𝐴 = -𝑒 𝐵 → 𝐴 = 𝐵 ) )
6		xnegeq	⊢ ( 𝐴 = 𝐵 → -𝑒 𝐴 = -𝑒 𝐵 )
7	5 6	impbid1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( -𝑒 𝐴 = -𝑒 𝐵 ↔ 𝐴 = 𝐵 ) )