Theorem eqeqan12d

Description: A useful inference for substituting definitions into an equality. See also eqeqan12dALT . (Contributed by NM, 9-Aug-1994) (Proof shortened by Andrew Salmon, 25-May-2011) Shorten other proofs. (Revised by Wolf Lammen, 23-Oct-2024)

	Ref	Expression
Hypotheses	eqeqan12d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	eqeqan12d.2	⊢ ( 𝜓 → 𝐶 = 𝐷 )
Assertion	eqeqan12d	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		eqeqan12d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		eqeqan12d.2	⊢ ( 𝜓 → 𝐶 = 𝐷 )
3	1	eqeq1d	⊢ ( 𝜑 → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐶 ) )
4	2	eqeq2d	⊢ ( 𝜓 → ( 𝐵 = 𝐶 ↔ 𝐵 = 𝐷 ) )
5	3 4	sylan9bb	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐷 ) )