Theorem bj-equsexvwd

Description: Variant of equsexvw . (Contributed by BJ, 7-Oct-2024)

	Ref	Expression
Hypotheses	bj-equsexvwd.nf0	$$⊢\phi \to \forall x\phi$$
	bj-equsexvwd.nf	$$⊢\phi \to Ⅎ\text{'}x\chi$$
	bj-equsexvwd.is	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	bj-equsexvwd	$$⊢\phi \to \left(\exists x\left(x=y\wedge \psi \right)\leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-equsexvwd.nf0	$$⊢\phi \to \forall x\phi$$
2		bj-equsexvwd.nf	$$⊢\phi \to Ⅎ\text{'}x\chi$$
3		bj-equsexvwd.is	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
4		alinexa	$$⊢\forall x\left(x=y\to \neg \psi \right)\leftrightarrow \neg \exists x\left(x=y\wedge \psi \right)$$
5		bj-nnfnt	$$⊢Ⅎ\text{'}x\chi \leftrightarrow Ⅎ\text{'}x\neg \chi$$
6	2 5	sylib	$$⊢\phi \to Ⅎ\text{'}x\neg \chi$$
7	3	notbid	$$⊢\left(\phi \wedge x=y\right)\to \left(\neg \psi \leftrightarrow \neg \chi \right)$$
8	1 6 7	bj-equsalvwd	$$⊢\phi \to \left(\forall x\left(x=y\to \neg \psi \right)\leftrightarrow \neg \chi \right)$$
9	4 8	bitr3id	$$⊢\phi \to \left(\neg \exists x\left(x=y\wedge \psi \right)\leftrightarrow \neg \chi \right)$$
10	9	con4bid	$$⊢\phi \to \left(\exists x\left(x=y\wedge \psi \right)\leftrightarrow \chi \right)$$