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Metamath Proof Explorer

Theorem reubidva

Description: Formula-building rule for restricted existential uniqueness quantifier (deduction form). (Contributed by NM, 13-Nov-2004) Reduce axiom usage. (Revised by Wolf Lammen, 14-Jan-2023)

		Ref	Expression
	Hypothesis	rmobidva.1	$⊢ (φ \land x \in A) \to (ψ \leftrightarrow χ)$
	Assertion	reubidva	$⊢ φ \to (\exists! x \in A ψ \leftrightarrow \exists! x \in A χ)$

Proof

Step	Hyp	Ref	Expression
1		rmobidva.1	$⊢ (φ \land x \in A) \to (ψ \leftrightarrow χ)$
2	1	pm5.32da	$⊢ φ \to ((x \in A \land ψ) \leftrightarrow (x \in A \land χ))$
3	2	eubidv	$⊢ φ \to (\exists! x (x \in A \land ψ) \leftrightarrow \exists! x (x \in A \land χ))$
4		df-reu	$⊢ \exists! x \in A ψ \leftrightarrow \exists! x (x \in A \land ψ)$
5		df-reu	$⊢ \exists! x \in A χ \leftrightarrow \exists! x (x \in A \land χ)$
6	3 4 5	3bitr4g	$⊢ φ \to (\exists! x \in A ψ \leftrightarrow \exists! x \in A χ)$