Theorem rexsng

Description: Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012) Avoid ax-10 , ax-12 . (Revised by GG, 30-Sep-2024)

	Ref	Expression
Hypothesis	ralsng.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	rexsng	$$⊢A\in V\to \left(\exists x\in \left\{A\right\}\phi \leftrightarrow \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ralsng.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
2	1	notbid	$$⊢x=A\to \left(\neg \phi \leftrightarrow \neg \psi \right)$$
3	2	ralsng	$$⊢A\in V\to \left(\forall x\in \left\{A\right\}\neg \phi \leftrightarrow \neg \psi \right)$$
4		dfrex2	$$⊢\exists x\in \left\{A\right\}\phi \leftrightarrow \neg \forall x\in \left\{A\right\}\neg \phi$$
5		bicom1	$$⊢\left(\forall x\in \left\{A\right\}\neg \phi \leftrightarrow \neg \psi \right)\to \left(\neg \psi \leftrightarrow \forall x\in \left\{A\right\}\neg \phi \right)$$
6	5	con1bid	$$⊢\left(\forall x\in \left\{A\right\}\neg \phi \leftrightarrow \neg \psi \right)\to \left(\neg \forall x\in \left\{A\right\}\neg \phi \leftrightarrow \psi \right)$$
7	4 6	bitrid	$$⊢\left(\forall x\in \left\{A\right\}\neg \phi \leftrightarrow \neg \psi \right)\to \left(\exists x\in \left\{A\right\}\phi \leftrightarrow \psi \right)$$
8	3 7	syl	$$⊢A\in V\to \left(\exists x\in \left\{A\right\}\phi \leftrightarrow \psi \right)$$