Theorem r19.21bi

Description: Inference from Theorem 19.21 of Margaris p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994) (Proof shortened by Wolf Lammen, 11-Jun-2023)

	Ref	Expression
Hypothesis	r19.21bi.1	$$⊢\phi \to \forall x\in A\psi$$
Assertion	r19.21bi	$$⊢\left(\phi \wedge x\in A\right)\to \psi$$

Proof

Step	Hyp	Ref	Expression
1		r19.21bi.1	$$⊢\phi \to \forall x\in A\psi$$
2		rspa	$$⊢\left(\forall x\in A\psi \wedge x\in A\right)\to \psi$$
3	1 2	sylan	$$⊢\left(\phi \wedge x\in A\right)\to \psi$$