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Metamath Proof Explorer

Theorem rabimbieq

Description: Restricted equivalent wff's correspond to restricted class abstractions which are equal with the same class. (Contributed by Peter Mazsa, 22-Jul-2021)

	Ref	Expression
Hypotheses	rabimbieq.1	$⊢ B = \{x \in A \| φ\}$
	rabimbieq.2	$⊢ x \in A \to (φ \leftrightarrow ψ)$
Assertion	rabimbieq	$⊢ B = \{x \in A \| ψ\}$

Proof

Step	Hyp	Ref	Expression
1		rabimbieq.1	$⊢ B = \{x \in A \| φ\}$
2		rabimbieq.2	$⊢ x \in A \to (φ \leftrightarrow ψ)$
3	2	rabbiia	$⊢ \{x \in A \| φ\} = \{x \in A \| ψ\}$
4	1 3	eqtri	$⊢ B = \{x \in A \| ψ\}$