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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	$⊢ φ \to A = B$
	eqeq12d.2	$⊢ φ \to C = D$
Assertion	eqeq12d	$⊢ φ \to (A = C \leftrightarrow B = D)$

Proof

Step	Hyp	Ref	Expression
1		eqeq12d.1	$⊢ φ \to A = B$
2		eqeq12d.2	$⊢ φ \to C = D$
3	1 2	eqeqan12d	$⊢ (φ \land φ) \to (A = C \leftrightarrow B = D)$
4	3	anidms	$⊢ φ \to (A = C \leftrightarrow B = D)$