Theorem ralprgf

Description: Convert a restricted universal quantification over a pair to a conjunction, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 17-Sep-2011) (Revised by AV, 8-Apr-2023)

	Ref	Expression
Hypotheses	ralprgf.1	$$⊢Ⅎx\psi$$
	ralprgf.2	$$⊢Ⅎx\chi$$
	ralprgf.a	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
	ralprgf.b	$$⊢x=B\to \left(\phi \leftrightarrow \chi \right)$$
Assertion	ralprgf	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\forall x\in \left\{A,B\right\}\phi \leftrightarrow \left(\psi \wedge \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ralprgf.1	$$⊢Ⅎx\psi$$
2		ralprgf.2	$$⊢Ⅎx\chi$$
3		ralprgf.a	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
4		ralprgf.b	$$⊢x=B\to \left(\phi \leftrightarrow \chi \right)$$
5		df-pr	$$⊢\left\{A,B\right\}=\left\{A\right\}\cup \left\{B\right\}$$
6	5	raleqi	$$⊢\forall x\in \left\{A,B\right\}\phi \leftrightarrow \forall x\in \left(\left\{A\right\}\cup \left\{B\right\}\right)\phi$$
7		ralunb	$$⊢\forall x\in \left(\left\{A\right\}\cup \left\{B\right\}\right)\phi \leftrightarrow \left(\forall x\in \left\{A\right\}\phi \wedge \forall x\in \left\{B\right\}\phi \right)$$
8	6 7	bitri	$$⊢\forall x\in \left\{A,B\right\}\phi \leftrightarrow \left(\forall x\in \left\{A\right\}\phi \wedge \forall x\in \left\{B\right\}\phi \right)$$
9	1 3	ralsngf	$$⊢A\in V\to \left(\forall x\in \left\{A\right\}\phi \leftrightarrow \psi \right)$$
10	2 4	ralsngf	$$⊢B\in W\to \left(\forall x\in \left\{B\right\}\phi \leftrightarrow \chi \right)$$
11	9 10	bi2anan9	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\left(\forall x\in \left\{A\right\}\phi \wedge \forall x\in \left\{B\right\}\phi \right)\leftrightarrow \left(\psi \wedge \chi \right)\right)$$
12	8 11	bitrid	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\forall x\in \left\{A,B\right\}\phi \leftrightarrow \left(\psi \wedge \chi \right)\right)$$