Theorem opnneilem

Description: Lemma factoring out common proof steps of opnneil and opnneirv . (Contributed by Zhi Wang, 31-Aug-2024)

	Ref	Expression
Hypothesis	opnneilem.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	opnneilem	$$⊢\phi \to \left(\exists x\in J\left(S\subseteq x\wedge \psi \right)\leftrightarrow \exists y\in J\left(S\subseteq y\wedge \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		opnneilem.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
2		sseq2	$$⊢x=y\to \left(S\subseteq x\leftrightarrow S\subseteq y\right)$$
3	2	adantl	$$⊢\left(\phi \wedge x=y\right)\to \left(S\subseteq x\leftrightarrow S\subseteq y\right)$$
4	3 1	anbi12d	$$⊢\left(\phi \wedge x=y\right)\to \left(\left(S\subseteq x\wedge \psi \right)\leftrightarrow \left(S\subseteq y\wedge \chi \right)\right)$$
5	4	cbvrexdva	$$⊢\phi \to \left(\exists x\in J\left(S\subseteq x\wedge \psi \right)\leftrightarrow \exists y\in J\left(S\subseteq y\wedge \chi \right)\right)$$