Theorem r19.28z

Description: Restricted quantifier version of Theorem 19.28 of Margaris p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 26-Oct-2010)

	Ref	Expression
Hypothesis	r19.3rz.1	$$⊢Ⅎx\phi$$
Assertion	r19.28z	$$⊢A\ne \varnothing \to \left(\forall x\in A\left(\phi \wedge \psi \right)\leftrightarrow \left(\phi \wedge \forall x\in A\psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		r19.3rz.1	$$⊢Ⅎx\phi$$
2		r19.26	$$⊢\forall x\in A\left(\phi \wedge \psi \right)\leftrightarrow \left(\forall x\in A\phi \wedge \forall x\in A\psi \right)$$
3	1	r19.3rz	$$⊢A\ne \varnothing \to \left(\phi \leftrightarrow \forall x\in A\phi \right)$$
4	3	anbi1d	$$⊢A\ne \varnothing \to \left(\left(\phi \wedge \forall x\in A\psi \right)\leftrightarrow \left(\forall x\in A\phi \wedge \forall x\in A\psi \right)\right)$$
5	2 4	bitr4id	$$⊢A\ne \varnothing \to \left(\forall x\in A\left(\phi \wedge \psi \right)\leftrightarrow \left(\phi \wedge \forall x\in A\psi \right)\right)$$