Theorem ra4

Description: Restricted quantifier version of Axiom 5 of Mendelson p. 69. This is the axiom stdpc5 of standard predicate calculus for a restricted domain. See ra4v for a version requiring fewer axioms. (Contributed by NM, 16-Jan-2004) (Proof shortened by BJ, 27-Mar-2020)

	Ref	Expression
Hypothesis	ra4.1	$$⊢Ⅎx\phi$$
Assertion	ra4	$$⊢\forall x\in A\left(\phi \to \psi \right)\to \left(\phi \to \forall x\in A\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ra4.1	$$⊢Ⅎx\phi$$
2	1	r19.21	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\in A\psi \right)$$
3	2	biimpi	$$⊢\forall x\in A\left(\phi \to \psi \right)\to \left(\phi \to \forall x\in A\psi \right)$$