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Metamath Proof Explorer

Theorem breqan12d

Description: Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996)

	Ref	Expression
Hypotheses	breq1d.1	$⊢ φ \to A = B$
	breqan12i.2	$⊢ ψ \to C = D$
Assertion	breqan12d	$⊢ (φ \land ψ) \to (A R C \leftrightarrow B R D)$

Step	Hyp	Ref	Expression
1		breq1d.1	$⊢ φ \to A = B$
2		breqan12i.2	$⊢ ψ \to C = D$
3		breq12	$⊢ (A = B \land C = D) \to (A R C \leftrightarrow B R D)$
4	1 2 3	syl2an	$⊢ (φ \land ψ) \to (A R C \leftrightarrow B R D)$