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Metamath Proof Explorer

Theorem rmobidva

Description: Formula-building rule for restricted at-most-one quantifier (deduction form). (Contributed by NM, 16-Jun-2017) Avoid ax-6 , ax-7 , ax-12 . (Revised by Wolf Lammen, 23-Nov-2024)

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

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	mobidv	$⊢ φ \to (\exists^{} x (x \in A \land ψ) \leftrightarrow \exists^{} x (x \in A \land χ))$
4		df-rmo	$⊢ \exists^{} x \in A ψ \leftrightarrow \exists^{} x (x \in A \land ψ)$
5		df-rmo	$⊢ \exists^{} x \in A χ \leftrightarrow \exists^{} x (x \in A \land χ)$
6	3 4 5	3bitr4g	$⊢ φ \to (\exists^{} x \in A ψ \leftrightarrow \exists^{} x \in A χ)$