Theorem rexprg

Description: Convert a restricted existential quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011) (Revised by Mario Carneiro, 23-Apr-2015) (Proof shortened by AV, 8-Apr-2023)

	Ref	Expression
Hypotheses	ralprg.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
	ralprg.2	$$⊢x=B\to \left(\phi \leftrightarrow \chi \right)$$
Assertion	rexprg	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\exists x\in \left\{A,B\right\}\phi \leftrightarrow \left(\psi \vee \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ralprg.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
2		ralprg.2	$$⊢x=B\to \left(\phi \leftrightarrow \chi \right)$$
3		nfv	$$⊢Ⅎx\psi$$
4		nfv	$$⊢Ⅎx\chi$$
5	3 4 1 2	rexprgf	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\exists x\in \left\{A,B\right\}\phi \leftrightarrow \left(\psi \vee \chi \right)\right)$$