Theorem eqeq2d

Description: Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993) Allow shortening of eqeq2 . (Revised by Wolf Lammen, 19-Nov-2019)

	Ref	Expression
Hypothesis	eqeq2d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	eqeq2d	⊢ ( 𝜑 → ( 𝐶 = 𝐴 ↔ 𝐶 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		eqeq2d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2	1	eqeq1d	⊢ ( 𝜑 → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐶 ) )
3		eqcom	⊢ ( 𝐶 = 𝐴 ↔ 𝐴 = 𝐶 )
4		eqcom	⊢ ( 𝐶 = 𝐵 ↔ 𝐵 = 𝐶 )
5	2 3 4	3bitr4g	⊢ ( 𝜑 → ( 𝐶 = 𝐴 ↔ 𝐶 = 𝐵 ) )