Theorem ralunsn

Description: Restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012) (Revised by Mario Carneiro, 23-Apr-2015)

	Ref	Expression
Hypothesis	ralunsn.1	$$⊢x=B\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	ralunsn	$$⊢B\in C\to \left(\forall x\in \left(A\cup \left\{B\right\}\right)\phi \leftrightarrow \left(\forall x\in A\phi \wedge \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ralunsn.1	$$⊢x=B\to \left(\phi \leftrightarrow \psi \right)$$
2		ralunb	$$⊢\forall x\in \left(A\cup \left\{B\right\}\right)\phi \leftrightarrow \left(\forall x\in A\phi \wedge \forall x\in \left\{B\right\}\phi \right)$$
3	1	ralsng	$$⊢B\in C\to \left(\forall x\in \left\{B\right\}\phi \leftrightarrow \psi \right)$$
4	3	anbi2d	$$⊢B\in C\to \left(\left(\forall x\in A\phi \wedge \forall x\in \left\{B\right\}\phi \right)\leftrightarrow \left(\forall x\in A\phi \wedge \psi \right)\right)$$
5	2 4	bitrid	$$⊢B\in C\to \left(\forall x\in \left(A\cup \left\{B\right\}\right)\phi \leftrightarrow \left(\forall x\in A\phi \wedge \psi \right)\right)$$