Theorem equsal

Description: An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . See equsalvw and equsalv for versions with disjoint variable conditions proved from fewer axioms. See also the dual form equsex . (Contributed by NM, 2-Jun-1993) (Proof shortened by Andrew Salmon, 12-Aug-2011) (Revised by Mario Carneiro, 3-Oct-2016) (Proof shortened by Wolf Lammen, 5-Feb-2018) (New usage is discouraged.)

	Ref	Expression
Hypotheses	equsal.1	$$⊢Ⅎx\psi$$
	equsal.2	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	equsal	$$⊢\forall x\left(x=y\to \phi \right)\leftrightarrow \psi$$

Proof

Step	Hyp	Ref	Expression
1		equsal.1	$$⊢Ⅎx\psi$$
2		equsal.2	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
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		ax6e	$$⊢\exists xx=y$$
7	6	a1bi	$$⊢\psi \leftrightarrow \left(\exists xx=y\to \psi \right)$$
8	3 5 7	3bitr4i	$$⊢\forall x\left(x=y\to \phi \right)\leftrightarrow \psi$$