rspceb2dv

Description: Restricted existential specialization, using implicit substitution in both directions. (Contributed by Zhi Wang, 28-Sep-2024)

	Ref	Expression
Hypotheses	rspceb2dv.1	$⊢ (φ \land x \in B) \to (ψ \to χ)$
	rspceb2dv.2	$⊢ (φ \land χ) \to A \in B$
	rspceb2dv.3	$⊢ (φ \land χ) \to θ$
	rspceb2dv.4	$⊢ x = A \to (ψ \leftrightarrow θ)$
Assertion	rspceb2dv	$⊢ φ \to (\exists x \in B ψ \leftrightarrow χ)$

Step	Hyp	Ref	Expression
1		rspceb2dv.1	$⊢ (φ \land x \in B) \to (ψ \to χ)$
2		rspceb2dv.2	$⊢ (φ \land χ) \to A \in B$
3		rspceb2dv.3	$⊢ (φ \land χ) \to θ$
4		rspceb2dv.4	$⊢ x = A \to (ψ \leftrightarrow θ)$
5	1	rexlimdva	$⊢ φ \to (\exists x \in B ψ \to χ)$
6	4	rspcev	$⊢ (A \in B \land θ) \to \exists x \in B ψ$
7	2 3 6	syl2anc	$⊢ (φ \land χ) \to \exists x \in B ψ$
8	7	ex	$⊢ φ \to (χ \to \exists x \in B ψ)$
9	5 8	impbid	$⊢ φ \to (\exists x \in B ψ \leftrightarrow χ)$

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