Theorem ralpr

Description: Convert a restricted universal quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007) (Revised by Mario Carneiro, 23-Apr-2015)

	Ref	Expression
Hypotheses	ralpr.1	$$⊢A\in \mathrm{V}$$
	ralpr.2	$$⊢B\in \mathrm{V}$$
	ralpr.3	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
	ralpr.4	$$⊢x=B\to \left(\phi \leftrightarrow \chi \right)$$
Assertion	ralpr	$$⊢\forall x\in \left\{A,B\right\}\phi \leftrightarrow \left(\psi \wedge \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ralpr.1	$$⊢A\in \mathrm{V}$$
2		ralpr.2	$$⊢B\in \mathrm{V}$$
3		ralpr.3	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
4		ralpr.4	$$⊢x=B\to \left(\phi \leftrightarrow \chi \right)$$
5	3 4	ralprg	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to \left(\forall x\in \left\{A,B\right\}\phi \leftrightarrow \left(\psi \wedge \chi \right)\right)$$
6	1 2 5	mp2an	$$⊢\forall x\in \left\{A,B\right\}\phi \leftrightarrow \left(\psi \wedge \chi \right)$$