Theorem eqeq12d

Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993) (Proof shortened by Andrew Salmon, 25-May-2011)

	Ref	Expression
Hypotheses	eqeq12d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	eqeq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	eqeq12d	⊢ ( 𝜑 → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		eqeq12d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		eqeq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3		eqeq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐷 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐷 ) )