Theorem wl-rgenw

Description: Generalization rule for restricted quantification. (Contributed by Wolf Lammen, 10-Jun-2023)

	Ref	Expression
Hypothesis	wl-rgenw.1	$$⊢\phi$$
Assertion	wl-rgenw	$$⊢\forall x:A\phi$$

Proof

Step	Hyp	Ref	Expression
1		wl-rgenw.1	$$⊢\phi$$
2		wl-dfralsb	$$⊢\forall x:A\phi \leftrightarrow \forall z\left(z\in A\to \left[z/x\right]\phi \right)$$
3	1	sbt	$$⊢\left[z/x\right]\phi$$
4	3	a1i	$$⊢z\in A\to \left[z/x\right]\phi$$
5	2 4	mpgbir	$$⊢\forall x:A\phi$$