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Metamath Proof Explorer

Theorem rspcedvdw

Description: Version of rspcedvd where the implicit substitution hypothesis does not have an antecedent, which also avoids a disjoint variable condition on ph , x . (Contributed by SN, 20-Aug-2024)

	Ref	Expression
Hypotheses	rspcedvdw.s	$⊢ x = A \to (ψ \leftrightarrow χ)$
	rspcedvdw.1	$⊢ φ \to A \in B$
	rspcedvdw.2	$⊢ φ \to χ$
Assertion	rspcedvdw	$⊢ φ \to \exists x \in B ψ$

Proof

Step	Hyp	Ref	Expression
1		rspcedvdw.s	$⊢ x = A \to (ψ \leftrightarrow χ)$
2		rspcedvdw.1	$⊢ φ \to A \in B$
3		rspcedvdw.2	$⊢ φ \to χ$
4	1	rspcev	$⊢ (A \in B \land χ) \to \exists x \in B ψ$
5	2 3 4	syl2anc	$⊢ φ \to \exists x \in B ψ$