Theorem equs45f

Description: Two ways of expressing substitution when y is not free in ph . The implication "to the left" is equs4 and does not require the nonfreeness hypothesis. Theorem sbalex replaces the nonfreeness hypothesis with a disjoint variable condition and equs5 replaces it with a distinctor antecedent. (Contributed by NM, 25-Apr-2008) (Revised by Mario Carneiro, 4-Oct-2016) Usage of this theorem is discouraged because it depends on ax-13 . Use sbalex instead. (New usage is discouraged.)

	Ref	Expression
Hypothesis	equs45f.1	$$⊢Ⅎy\phi$$
Assertion	equs45f	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \forall x\left(x=y\to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		equs45f.1	$$⊢Ⅎy\phi$$
2	1	nf5ri	$$⊢\phi \to \forall y\phi$$
3	2	anim2i	$$⊢\left(x=y\wedge \phi \right)\to \left(x=y\wedge \forall y\phi \right)$$
4	3	eximi	$$⊢\exists x\left(x=y\wedge \phi \right)\to \exists x\left(x=y\wedge \forall y\phi \right)$$
5		equs5a	$$⊢\exists x\left(x=y\wedge \forall y\phi \right)\to \forall x\left(x=y\to \phi \right)$$
6	4 5	syl	$$⊢\exists x\left(x=y\wedge \phi \right)\to \forall x\left(x=y\to \phi \right)$$
7		equs4	$$⊢\forall x\left(x=y\to \phi \right)\to \exists x\left(x=y\wedge \phi \right)$$
8	6 7	impbii	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \forall x\left(x=y\to \phi \right)$$