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Metamath Proof Explorer

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) (Proof shortened by Wolf Lammen, 23-Oct-2024)

	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	1 2	eqeqan12d	⊢ ( ( 𝜑 ∧ 𝜑 ) → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐷 ) )
4	3	anidms	⊢ ( 𝜑 → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐷 ) )