Metamath Proof Explorer

Theorem ralbidb

Description: Formula-building rule for restricted universal quantifier and additional condition (deduction form). See ralbidc for a more generalized form. (Contributed by Zhi Wang, 6-Sep-2024)

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

Proof

Step	Hyp	Ref	Expression
1		ralbidb.1	$⊢ φ \to (x \in A \leftrightarrow (x \in B \land ψ))$
2		ralbidb.2	$⊢ (φ \land x \in A) \to (χ \leftrightarrow θ)$
3	1 2	logic1a	$⊢ φ \to ((x \in A \to χ) \leftrightarrow ((x \in B \land ψ) \to θ))$
4		impexp	$⊢ ((x \in B \land ψ) \to θ) \leftrightarrow (x \in B \to (ψ \to θ))$
5	3 4	bitrdi	$⊢ φ \to ((x \in A \to χ) \leftrightarrow (x \in B \to (ψ \to θ)))$
6	5	ralbidv2	$⊢ φ \to (\forall x \in A χ \leftrightarrow \forall x \in B (ψ \to θ))$