Theorem equsalv

Description: An equivalence related to implicit substitution. Version of equsal with a disjoint variable condition, which does not require ax-13 . See equsalvw for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsexv . (Contributed by NM, 2-Jun-1993) (Revised by BJ, 31-May-2019)

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

Proof

Step	Hyp	Ref	Expression
1		equsalv.nf	$$⊢Ⅎx\psi$$
2		equsalv.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
3	1	19.23	$$⊢\forall x\left(x=y\to \psi \right)\leftrightarrow \left(\exists xx=y\to \psi \right)$$
4	2	pm5.74i	$$⊢\left(x=y\to \phi \right)\leftrightarrow \left(x=y\to \psi \right)$$
5	4	albii	$$⊢\forall x\left(x=y\to \phi \right)\leftrightarrow \forall x\left(x=y\to \psi \right)$$
6		ax6ev	$$⊢\exists xx=y$$
7	6	a1bi	$$⊢\psi \leftrightarrow \left(\exists xx=y\to \psi \right)$$
8	3 5 7	3bitr4i	$$⊢\forall x\left(x=y\to \phi \right)\leftrightarrow \psi$$