Theorem bj-equsalhv

Description: Version of equsalh with a disjoint variable condition, which does not require ax-13 . Remark: this is the same as equsalhw . TODO: delete after moving the following paragraph somewhere.

Remarks: equsexvw has been moved to Main; Theorem ax13lem2 has a DV version which is a simple consequence of ax5e ; Theorems nfeqf2 , dveeq2 , nfeqf1 , dveeq1 , nfeqf , axc9 , ax13 , have dv versions which are simple consequences of ax-5 . (Contributed by BJ, 14-Jun-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bj-equsalhv.nf	$$⊢\psi \to \forall x\psi$$
	bj-equsalhv.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	bj-equsalhv	$$⊢\forall x\left(x=y\to \phi \right)\leftrightarrow \psi$$

Proof

Step	Hyp	Ref	Expression
1		bj-equsalhv.nf	$$⊢\psi \to \forall x\psi$$
2		bj-equsalhv.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
3	1	nf5i	$$⊢Ⅎx\psi$$
4	3 2	equsalv	$$⊢\forall x\left(x=y\to \phi \right)\leftrightarrow \psi$$