Theorem equsex

Description: An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . See equsexvw and equsexv for versions with disjoint variable conditions proved from fewer axioms. See also the dual form equsal . See equsexALT for an alternate proof. (Contributed by NM, 5-Aug-1993) (Revised by Mario Carneiro, 3-Oct-2016) (Proof shortened by Wolf Lammen, 6-Feb-2018) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		equsal.1	$$⊢Ⅎx\psi$$
2		equsal.2	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
3	2	biimpa	$$⊢\left(x=y\wedge \phi \right)\to \psi$$
4	1 3	exlimi	$$⊢\exists x\left(x=y\wedge \phi \right)\to \psi$$
5	1 2	equsal	$$⊢\forall x\left(x=y\to \phi \right)\leftrightarrow \psi$$
6		equs4	$$⊢\forall x\left(x=y\to \phi \right)\to \exists x\left(x=y\wedge \phi \right)$$
7	5 6	sylbir	$$⊢\psi \to \exists x\left(x=y\wedge \phi \right)$$
8	4 7	impbii	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \psi$$