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Metamath Proof Explorer

Theorem rspcdv2

Description: Restricted specialization, using implicit substitution. (Contributed by Stanislas Polu, 9-Mar-2020)

	Ref	Expression
Hypotheses	rspcdv2.1	$⊢ (φ \land x = A) \to (ψ \leftrightarrow χ)$
	rspcdv2.2	$⊢ φ \to A \in B$
	rspcdv2.3	$⊢ φ \to \forall x \in B ψ$
Assertion	rspcdv2	$⊢ φ \to χ$

Step	Hyp	Ref	Expression
1		rspcdv2.1	$⊢ (φ \land x = A) \to (ψ \leftrightarrow χ)$
2		rspcdv2.2	$⊢ φ \to A \in B$
3		rspcdv2.3	$⊢ φ \to \forall x \in B ψ$
4	2 1	rspcdv	$⊢ φ \to (\forall x \in B ψ \to χ)$
5	3 4	mpd	$⊢ φ \to χ$