Theorem eqeqan12rd

Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994)

	Ref	Expression
Hypotheses	eqeqan12rd.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	eqeqan12rd.2	⊢ ( 𝜓 → 𝐶 = 𝐷 )
Assertion	eqeqan12rd	⊢ ( ( 𝜓 ∧ 𝜑 ) → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		eqeqan12rd.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		eqeqan12rd.2	⊢ ( 𝜓 → 𝐶 = 𝐷 )
3	1 2	eqeqan12d	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐷 ) )
4	3	ancoms	⊢ ( ( 𝜓 ∧ 𝜑 ) → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐷 ) )