Theorem r19.12sn

Description: Special case of r19.12 where its converse holds. (Contributed by NM, 19-May-2008) (Revised by Mario Carneiro, 23-Apr-2015) (Revised by BJ, 18-Mar-2020)

	Ref	Expression
Assertion	r19.12sn	$$⊢A\in V\to \left(\exists x\in \left\{A\right\}\forall y\in B\phi \leftrightarrow \forall y\in B\exists x\in \left\{A\right\}\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sbcralg	$$⊢A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\forall y\in B\phi \leftrightarrow \forall y\in B\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$
2		rexsns	$$⊢\exists x\in \left\{A\right\}\forall y\in B\phi \leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\forall y\in B\phi$$
3		rexsns	$$⊢\exists x\in \left\{A\right\}\phi \leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi$$
4	3	ralbii	$$⊢\forall y\in B\exists x\in \left\{A\right\}\phi \leftrightarrow \forall y\in B\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi$$
5	1 2 4	3bitr4g	$$⊢A\in V\to \left(\exists x\in \left\{A\right\}\forall y\in B\phi \leftrightarrow \forall y\in B\exists x\in \left\{A\right\}\phi \right)$$