Theorem rspn0

Description: Specialization for restricted generalization with a nonempty class. (Contributed by Alexander van der Vekens, 6-Sep-2018) Avoid ax-10 , ax-12 . (Revised by Gino Giotto, 28-Jun-2024)

	Ref	Expression
Assertion	rspn0	$$⊢A\ne \varnothing \to \left(\forall x\in A\phi \to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		n0	$$⊢A\ne \varnothing \leftrightarrow \exists xx\in A$$
2		df-ral	$$⊢\forall x\in A\phi \leftrightarrow \forall x\left(x\in A\to \phi \right)$$
3		exim	$$⊢\forall x\left(x\in A\to \phi \right)\to \left(\exists xx\in A\to \exists x\phi \right)$$
4		ax5e	$$⊢\exists x\phi \to \phi$$
5	3 4	syl6com	$$⊢\exists xx\in A\to \left(\forall x\left(x\in A\to \phi \right)\to \phi \right)$$
6	2 5	biimtrid	$$⊢\exists xx\in A\to \left(\forall x\in A\phi \to \phi \right)$$
7	1 6	sylbi	$$⊢A\ne \varnothing \to \left(\forall x\in A\phi \to \phi \right)$$