Theorem 19.8a

Description: If a wff is true, it is true for at least one instance. Special case of Theorem 19.8 of Margaris p. 89. See 19.8v for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 9-Jan-1993) Allow a shortening of sp . (Revised by Wolf Lammen, 13-Jan-2018) (Proof shortened by Wolf Lammen, 8-Dec-2019)

	Ref	Expression
Assertion	19.8a	$$⊢\phi \to \exists x\phi$$

Proof

Step	Hyp	Ref	Expression
1		ax12v	$$⊢x=y\to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)$$
2		alequexv	$$⊢\forall x\left(x=y\to \phi \right)\to \exists x\phi$$
3	1 2	syl6	$$⊢x=y\to \left(\phi \to \exists x\phi \right)$$
4		ax6evr	$$⊢\exists yx=y$$
5	3 4	exlimiiv	$$⊢\phi \to \exists x\phi$$