Theorem bj-equsexval

Description: Special case of equsexv proved from core axioms, ax-10 (modal5), and hba1 (modal4). (Contributed by BJ, 29-Dec-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	bj-equsexval.1	$$⊢x=y\to \left(\phi \leftrightarrow \forall x\psi \right)$$
Assertion	bj-equsexval	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \forall x\psi$$

Proof

Step	Hyp	Ref	Expression
1		bj-equsexval.1	$$⊢x=y\to \left(\phi \leftrightarrow \forall x\psi \right)$$
2	1	pm5.32i	$$⊢\left(x=y\wedge \phi \right)\leftrightarrow \left(x=y\wedge \forall x\psi \right)$$
3	2	exbii	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \exists x\left(x=y\wedge \forall x\psi \right)$$
4		ax6ev	$$⊢\exists xx=y$$
5		bj-19.41al	$$⊢\exists x\left(x=y\wedge \forall x\psi \right)\leftrightarrow \left(\exists xx=y\wedge \forall x\psi \right)$$
6	4 5	mpbiran	$$⊢\exists x\left(x=y\wedge \forall x\psi \right)\leftrightarrow \forall x\psi$$
7	3 6	bitri	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \forall x\psi$$